Chapter 9 Gases Section 2 Relating Pressure, Volume, Amount, and Temperature: The Ideal Gas Law

If the number of moles of an ideal gas are kept constant under two different sets of conditions, a useful mathematical relationship called the combined gas law is obtained:

(𝑃1𝑉1)/𝑇1 = (𝑃2𝑉2)/𝑇2 using units of atm, L, and K. Both sets of conditions are equal to the product of n × R (where n = the number of moles of the gas and R is the ideal gas law constant)

The units used to express pressure, volume, and temperature will determine the proper form of the gas constant as required by dimensional analysis, the most commonly encountered values being

0.08206 L atm mol^-1 K^-1 and 8.314 kPa L mol^-1 K^-1

Four separate laws have been discussed that relate pressure, volume, temperature, and the number of moles of the gas:

1.) Boyle's law: PV = constant at constant T and n 2.) Amontons's law: 𝑃/𝑇 = constant at constant V and n 3.) Charles's law: 𝑉/𝑇 = constant at constant P and n 4.) Avogadro's law: 𝑉/𝑛 = constant at constant P and T Combining these four laws yields the ideal gas law, a relation between the pressure, volume, temperature, and number of moles of a gas: 𝑃𝑉=𝑛𝑅𝑇 where P is the pressure of a gas, V is its volume, n is the number of moles of the gas, T is its temperature on the kelvin scale, and R is a constant called the ideal gas constant or the universal gas constant.

Check Your Learning A sample of oxygen, O2, occupies 32.2 mL at 30 °C and 452 torr. What volume will it occupy at -70 °C and the same pressure?

21.6mL

standard temperature and pressure (STP) for reporting properties of gases:

273.15 K and 1 atm (101.325 kPa).1 At STP, one mole of an ideal gas has a volume of about 22.4 L—this is referred to as the standard molar volume

Check Your Learning A sample of nitrogen, N2, occupies 45.0 mL at 27 °C and 600 torr. What pressure will it have if cooled to -73 °C while the volume remains constant?

400 torr

Check Your Learning What is the volume of a sample of ethane at 467 K and 1.1 atm if it occupies 405 mL at 298 K and 1.1 atm?

635mL

Because of this, the P- T relationship for gases is known as either

Amontons's law or Gay-Lussac's law

The relationship between the volume and temperature of a given amount of gas at constant pressure is known as

Charles's law in recognition of the French scientist and balloon flight pioneer Jacques Alexandre César Charles.

Avogadro's Law

For a confined gas, the volume (V) and number of moles (n) are directly proportional if the pressure and temperature both remain constant In equation form, this is written as: 𝑉 ∝ 𝑛 or 𝑉 = 𝑘 × 𝑛 or 𝑉1/𝑛1=𝑉2/𝑛2 Mathematical relationships can also be determined for the other variable pairs, such as P versus n, and n versus T.

Gases whose properties of P, V, and T are accurately described by the ideal gas law (or the other gas laws) are said to exhibit ideal behavior or to approximate the traits of an

Ideal gas

Pressure and Temperature: Amontons's Law: For a confined, constant volume of gas, the ratio 𝑃/𝑇 is therefore constant (i.e., 𝑃/𝑇=𝑘)

If the gas is initially in "Condition 1" (with P = P1 and T = T1), and then changes to "Condition 2" (with P = P2 and T = T2) we have that 𝑃1/𝑇1=𝑘 and 𝑃2/𝑇2=𝑘, which reduces to 𝑃1/T1=𝑃2/𝑇2 that temperatures must be on the kelvin scale for any gas law calculations (0 on the kelvin scale and the lowest possible temperature is called absolute zero) *Also note that there are at least three ways we can describe how the pressure of a gas changes as its temperature changes: We can use a table of values, a graph, or a mathematical equation

Volume and Pressure: Boyle's Law Decreasing the volume of a contained gas will increase its pressure, and increasing its volume will decrease its pressure

In fact, if the volume increases by a certain factor, the pressure decreases by the same factor, and vice versa

Boyle's law: Unlike the P- T and V- T relationships, pressure and volume are not directly proportional to each other

Instead, P and V exhibit inverse proportionality: Increasing the pressure results in a decrease of the volume of the gas. Mathematically this can be written: 𝑃 α 1/𝑉 or𝑃 = 𝑘·1/𝑉 or 𝑃·𝑉= 𝑘 or 𝑃1/𝑉1=𝑃2/𝑉2 With k being a constant *Graphically, this relationship is shown by the straight line that results when plotting the inverse of the pressure (1/𝑃) versus the volume ( V), or the inverse of volume (1/𝑉) versus the pressure ( P)

Example: Predicting Change in Pressure with Temperature A can of hair spray is used until it is empty except for the propellant, isobutane gas. (a) On the can is the warning "Store only at temperatures below 120 °F (48.8 °C). Do not incinerate." Why? (b) The gas in the can is initially at 24 °C and 360 kPa, and the can has a volume of 350 mL. If the can is left in a car that reaches 50 °C on a hot day, what is the new pressure in the can?

Solution: (a) The can contains an amount of isobutane gas at a constant volume, so if the temperature is increased by heating, the pressure will increase proportionately. High temperature could lead to high pressure, causing the can to burst. (Also, isobutane is combustible, so incineration could cause the can to explode.) (b) We are looking for a pressure change due to a temperature change at constant volume, so we will use Amontons's/Gay-Lussac's law. Taking P1 and T1 as the initial values, T2 as the temperature where the pressure is unknown and P2 as the unknown pressure, and converting °C to K, we have: P1/T1 = P2/T2 which means that 360 kPa / 297 K = P2/ 323K Rearranging and solving gives: P2 = 360 kPa x 323 K / 297 K = 390 kPa

Example: Volume of a Gas Sample The sample of gas in Figure 9.13 has a volume of 15.0 mL at a pressure of 13.0 psi. Determine the pressure of the gas at a volume of 7.5 mL, using: (a) the P- V graph (b) the 1/𝑃 vs. V graph (c) the Boyle's law equation Comment on the likely accuracy of each method

Solution: (a) Estimating from the P- V graph gives a value for P somewhere around 27 psi. (b) Estimating from the 1/𝑃 versus V graph give a value of about 26 psi. (c) From Boyle's law, we know that the product of pressure and volume ( PV) for a given sample of gas at a constant temperature is always equal to the same value. Therefore we have P1V1 = k and P2V2 = k which means that P1V1 = P2V2. Using P1 and V1 as the known values 13.0 psi and 15.0 mL, P2 as the pressure at which the volume is unknown, and V2 as the unknown volume, we have: 𝑃1𝑉1=𝑃2𝑉2 or 13.0psi × 15.0mL=𝑃2 × 7.5mL Solving: 𝑃2 = (13.0psi×15.0mL)/7.5mL = 26psi It was more difficult to estimate well from the P- V graph, so (a) is likely more inaccurate than (b) or (c). The calculation will be as accurate as the equation and measurements allow

Example: Measuring Temperature with a Volume Change Temperature is sometimes measured with a gas thermometer by observing the change in the volume of the gas as the temperature changes at constant pressure. The hydrogen in a particular hydrogen gas thermometer has a volume of 150.0 cm3 when immersed in a mixture of ice and water (0.00 °C). When immersed in boiling liquid ammonia, the volume of the hydrogen, at the same pressure, is 131.7 cm3. Find the temperature of boiling ammonia on the kelvin and Celsius scales.

Solution: A volume change caused by a temperature change at constant pressure means we should use Charles's law. Taking V1 and T1 as the initial values, T2 as the temperature at which the volume is unknown and V2 as the unknown volume, and converting °C into K we have: 𝑉1/𝑇1 = 𝑉2/𝑇2which means that 150.0cm / 3273.15K = 131.7cm3/𝑇2 Rearrangement gives 𝑇2= (131.7cm3 × 273.15K) /150.0cm3 = 239.8K Subtracting 273.15 from 239.8 K, we find that the temperature of the boiling ammonia on the Celsius scale is -33.4 °C

Example (Charles's Law): Predicting Change in Volume with Temperature A sample of carbon dioxide, CO2, occupies 0.300 L at 10 °C and 750 torr. What volume will the gas have at 30 °C and 750 torr?

Solution: Because we are looking for the volume change caused by a temperature change at constant pressure, this is a job for Charles's law. Taking V1 and T1 as the initial values, T2 as the temperature at which the volume is unknown and V2 as the unknown volume, and converting °C into K we have: 𝑉1/𝑇1=𝑉2/𝑇2 which means that0.300L/ 283K= 𝑉2/ 303K Rearranging and solving gives: 𝑉2=(0.300L×303K)/283K=0.321L This answer supports our expectation from Charles's law, namely, that raising the gas temperature (from 283 K to 303 K) at a constant pressure will yield an increase in its volume (from 0.300 L to 0.321 L).

Example: Using the Combined Gas Law When filled with air, a typical scuba tank with a volume of 13.2 L has a pressure of 153 atm. If the water temperature is 27 °C, how many liters of air will such a tank provide to a diver's lungs at a depth of approximately 70 feet in the ocean where the pressure is 3.13 atm?

Solution: Letting 1 represent the air in the scuba tank and 2 represent the air in the lungs, and noting that body temperature (the temperature the air will be in the lungs) is 37 °C, we have: (𝑃1𝑉1)/𝑇1 = (𝑃2𝑉2)/𝑇2⟶[(153atm)(13.2L)]/(300K) = [(3.13atm)(𝑉2)]/(310K) Solving for V2: 𝑉2 = [(153atm)(13.2L)(310K)] / (300K)(3.13atm) = 667L *(Note: Be advised that this particular example is one in which the assumption of ideal gas behavior is not very reasonable, since it involves gases at relatively high pressures and low temperatures. Despite this limitation, the calculated volume can be viewed as a good "ballpark" estimate.)

Example: Using the Ideal Gas Law Methane, CH4, is being considered for use as an alternative automotive fuel to replace gasoline. One gallon of gasoline could be replaced by 655 g of CH4. What is the volume of this much methane at 25 °C and 745 torr?

Solution: We must rearrange PV = nRT to solve for V: 𝑉=𝑛𝑅𝑇/𝑃 If we choose to use R = 0.08206 L atm mol^-1 K^-1, then the amount must be in moles, temperature must be in kelvin, and pressure must be in atm. Converting into the "right" units: 𝑛 = 655gCH4 × 1mol/ 16.043g CH4 =40.8mol 𝑇 = 25°C + 273 = 298K 𝑃 = 745 torr × 1atm/ 760 torr = 0.980atm 𝑉 = 𝑛𝑅𝑇/𝑃 = [(40.8mol)(0.08206Latm mol^-1K^-1)(298K)] /0.980atm = 1.02 × 10^3L It would require 1020 L (269 gal) of gaseous methane at about 1 atm of pressure to replace 1 gal of gasoline. It requires a large container to hold enough methane at 1 atm to replace several gallons of gasoline

Amonton's Law/Gay-Lussac's Law

States that the pressure of a given amount of gas is directly proportional to its temperature on the kelvin scale when the volume is held constant Mathematically, this can be written: 𝑃 ∝ 𝑇 or𝑃 = constant x 𝑇 or 𝑃 = 𝑘 × 𝑇 *where ∝ means "is proportional to," and k is a proportionality constant that depends on the identity, amount, and volume of the gas.

Volume and Temperature: Charles's Law: These examples of the effect of temperature on the volume of a given amount of a confined gas at constant pressure are true in general:

The volume increases as the temperature increases, and decreases as the temperature decreases

Boyle's law

The volume of a given amount of gas held at constant temperature is inversely proportional to the pressure under which it is measured

Volume and Pressure: Boyle's Law If we partially fill an airtight syringe with air, the syringe contains a specific amount of air at constant temperature, say 25 °C. If we slowly push in the plunger while keeping temperature constant, the gas in the syringe is compressed into a smaller volume and its pressure increases; if we pull out the plunger, the volume increases and the pressure decreases

This example of the effect of volume on the pressure of a given amount of a confined gas is true in general

Breathing and Boyle's Law

What do you do about 20 times per minute for your whole life, without break, and often without even being aware of it? The answer, of course, is respiration, or breathing. How does it work? It turns out that the gas laws apply here. Your lungs take in gas that your body needs (oxygen) and get rid of waste gas (carbon dioxide). Lungs are made of spongy, stretchy tissue that expands and contracts while you breathe. When you inhale, your diaphragm and intercostal muscles (the muscles between your ribs) contract, expanding your chest cavity and making your lung volume larger. The increase in volume leads to a decrease in pressure (Boyle's law). This causes air to flow into the lungs (from high pressure to low pressure). When you exhale, the process reverses: Your diaphragm and rib muscles relax, your chest cavity contracts, and your lung volume decreases, causing the pressure to increase (Boyle's law again), and air flows out of the lungs (from high pressure to low pressure). You then breathe in and out again, and again, repeating this Boyle's law cycle for the rest of your life

ideal gas

a hypothetical gas that perfectly fits all the assumptions of the kinetic-molecular theory a hypothetical construct that real gases approximate under certain conditions

Volume and Temperature: Charles's Law: For a confined, constant pressure gas sample, 𝑉/𝑇 is constant (i.e., the ratio = k), and

as seen with the P- T relationship, this leads to another form of Charles's law: 𝑉1/𝑇1 = 𝑉2/𝑇2

Pressure and Temperature: Amontons's Law: Imagine filling a rigid container attached to a pressure gauge with gas and then sealing the container so that no gas may escape. If the container is cooled, the gas inside likewise gets colder and its pressure is observed to

decrease *Since the container is rigid and tightly sealed, both the volume and number of moles of gas remain constant If we heat the sphere, the gas inside gets hotter and the pressure increases

Boyle's law

decreasing the pressure of gases and increases the volume

Moles of Gas and Volume: Avogadro's Law The Italian scientist Amedeo Avogadro advanced a hypothesis in 1811 to account for the behavior of gases, stating that

equal volumes of all gases, measured under the same conditions of temperature and pressure, contain the same number of molecules. Over time, this relationship was supported by many experimental observations as expressed by Avogadro's law

Pressure and Temperature: Amontons's Law: This relationship between temperature and pressure is observed

for any sample of gas confined to a constant volume

An ideal gas is a

hypothetical construct that may be used along with kinetic molecular theory to effectively explain the gas laws

Pressure and Temperature: Amontons's Law: We find that temperature and pressure are linearly related, and if the temperature is on the kelvin scale, then P and T are directly proportional (again, when volume and moles of gas are held constant); if the temperature on the kelvin scale increases by a certain factor, the gas pressure

increases by the same factor

A number of scientists established the relationships between the macroscopic physical properties of gases, that is,

pressure, volume, temperature, and amount of gas

Pressure and Temperature: Amontons's Law: For a constant volume and amount of air, the pressure and temperature are directly proportional,

provided the temperature is in kelvin. (Measurements cannot be made at lower temperatures because of the condensation of the gas.)

Ideal Gas Law

relates gas quantities for gases and is quite accurate for low pressures and moderate temperatures *the relationship PV=nRT, which describes the behavior of an ideal gas

Volume and Temperature: Charles's Law: If we fill a balloon with air and seal it, the balloon contains a specific amount of air at atmospheric pressure, let's say 1 atm. If we put the balloon in a refrigerator, the gas inside gets cold and the balloon shrinks (although both the amount of gas and its pressure remain constant). If we make the balloon very cold, it will

shrink a great deal, and it expands again when it warms up

Charles's law states

that the volume of a given amount of gas is directly proportional to its temperature on the kelvin scale when the pressure is held constant Mathematically, this can be written as: 𝑉 α 𝑇 or 𝑉 = constant·𝑇 or 𝑉 = 𝑘·𝑇 or 𝑉1/𝑇1 = 𝑉2/𝑇2 *with k being a proportionality constant that depends on the amount and pressure of the gas.

The ideal gas equation contains five terms,

the gas constant R and the variable properties P, V, n, and T *Specifying any four of these terms will permit use of the ideal gas law to calculate the fifth term

The Ideal Gas Law

the relationship PV=nRT, which describes the behavior of an ideal gas

Guillaume Amontons was the first to empirically establish

the relationship between the pressure and the temperature of a gas (~1700), and Joseph Louis Gay-Lussac determined the relationship more precisely (~1800)