Molar heat capacities: A quantity of ideal gas requires 800 kJ to raise the temperature of the gas by 10.0 K when the gas is maintained at constant volume. The same quantity of gas requires 900 kJ to raise the temperature of the gas by 10.0 K when the gas is maintained at constant pressure. What is the adiabatic constant γ for this gas?

Question

Quizplus · Accepted Answer

The molar heat capacity at constant volume is given by $C_v=\frac{Q}{n\Delta T}$, where $Q$ is the amount of heat transferred, $n$ is the number of moles of gas, and $\Delta T$ is the change in temperature. Similarly, the molar heat capacity at constant pressure is given by $C_p=\frac{Q}{n\Delta T}$.

The adiabatic constant $\gamma$ is defined as the ratio of the molar heat capacities at constant pressure and constant volume: $$\gamma=\frac{C_p}{C_v}$$

Using the given values, we have:

$$C_v=\frac{800 	ext{ kJ}}{n	imes 10.0 	ext{ K}}=\frac{80}{n} 	ext{ J/K/mol}$$

$$C_p=\frac{900 	ext{ kJ}}{n	imes 10.0 	ext{ K}}=\frac{90}{n} 	ext{ J/K/mol}$$

Therefore,

$$\gamma=\frac{C_p}{C_v}=\frac{90/n}{80/n}=1.125$$

which is equal to $1+(\frac{2}{2}-\frac{1}{2})	imes\frac{f}{2}=1+\frac{f}{2}$ since the gas is monoatomic. Solving for $f$, we get:

$$f=2(\gamma-1)=2(1.125-1)=0.25$$

Therefore, since the gas is monoatomic, the answer is:

$$\gamma=1+\frac{f}{2}=1.125$$

The closest answer choice is B: 1.13.

Molar Heat Capacities: a Quantity of Ideal Gas Requires 800