10.2: Gas Laws

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Gas Laws

Exercise \(\PageIndex{1}\)

Use the ideal gas law to derive an equation that relates the remaining variables for a sample of an ideal gas if the following are held constant.

amount and volume
pressure and amount
temperature and volume
temperature and amount
pressure and temperature

Answer

P/T = constant
V/T = constant (Charles’ law)
P/n = constant
PV = constant (Boyle’s law)
V/n = constant (Avogadro’s law)

Ideal Gas Law

Exercise \(\PageIndex{2}\)

A sample of gaseous Cl₂ has a volume of 12.4 L at 500.0 K and 0.521 atm. How many moles are present?

Answer: \(n=\frac{PV}{RT}=\frac{0.521atm\left ( 12.4L \right )}{0.08206\frac{L\cdot atm}{mol\cdot K}\left ( 500K \right )}=0.157mol\)

Exercise \(\PageIndex{3}\)

What is the mass of chlorine gas in a 12.4L container at 500.0K and 0.521 atm?

Answer

\(n=\frac{PV}{RT}=\frac{0.521atm\left ( 12.4L \right )}{0.08206\frac{L\cdot atm}{mol\cdot K}\left ( 500K \right )}=0.157mol\)

\(0.157mol~Cl_{2}\left ( \frac{70.90g~Cl_{2}}{mol} \right )=11.1g\)

Exercise \(\PageIndex{4}\)

What volume is occupied by 1.06 mol of CO₂ gas at 299 K and a pressure of 0.89 atm?

Answer: \(V=\frac{nRT}{P}=\frac{1.06mol\left ( 0.08206\frac{L\cdot atm}{mol\cdot K} \right )299K}{0.89atm}=29L\)

Exercise \(\PageIndex{5}\)

What is the temperature of 1.41 mol of methane gas in a 5.0 L container at 1.00 atm?

Answer: \(T=\frac{PV}{nR}=\frac{1.00atm\left ( 5.0L \right )}{1.41mol\left ( 0.08206\frac{L\cdot atm}{mol\cdot K} \right )}=43K\)

Exercise \(\PageIndex{6}\)

What is the pressure exerted by a 1.75 mol sample of water at 7.0L and 20^oC?

Answer: \(P=\frac{nRT}{V}=\frac{1.75mol\left ( 0.08206\frac{L\cdot atm}{mol\cdot K} \right )293.15K}{7.0L}=5.8atm\)

Two State

Exercise \(\PageIndex{7}\)

What pressure would 6.1 mole of a gas in a rigid container have at 20.0 ^oC have if it had a pressure of 0.45 atm at -45.0 ^oC?

Answer

\(P_{1}=P_{2}\frac{T_{1}}{T_{2}}=\left ( 0.45atm \right )\left ( \frac{293.15K}{228.15K} \right )=0.58atm\)

Exercise \(\PageIndex{8}\)

A pressure tank containing chlorine gas at a pressure of 2.00 atm and a temperature of 40^oC is set with a pressure relief valve set to open at a pressure of 10.0 atm. At what temperature will the relief valve open?

Answer: \(T_{2}=T_{1}\frac{P_{2}}{P_{1}}= \left ( 313.15K \right )\left ( \frac{10.0atm}{2.00atm} \right )=1570K\)

Exercise \(\PageIndex{9}\)

A gas is in a sealed cylinder at 25^oC with a piston which can expand or contract to change the volume. The initial volume and pressure are 75.0 L and 980.0 torr. A force is applied to the piston and the volume adjusts to a final pressure and temperature of 5.00 atm at 188^oC. What is the final volume?

Answer: \(V_{2}=V_{1}\left ( \frac{P_{1}}{P_{2}} \right )\left ( \frac{T_{2}}{T_{1}} \right )=75.0L\left ( \frac{980torr\left ( \frac{1atm}{760torr} \right )}{5.00atm} \right )\left ( \frac{461.15K}{298K} \right )=29.9L\)