Question

An ideal gas with adiabatic exponent $\gamma$ is heated at a constant pressure. It absorbs total $Q$ amount of heat. Fraction of the heat used to increase its temperature of the gas is

• A. $\gamma$
• B. $\frac{1}{\gamma }$
• C. $1-\frac{1}{\gamma }$
• D. $2\gamma$

Answer 1

The correct option is B $\frac{1}{\gamma }$

Using first law of thermodynamics,
$\Delta Q=\Delta U+\Delta W$
Since the gas is heated at constant pressure we can say that,
Heat supplied to the gas at constant pressure $\Delta Q=n{C}_P\Delta T$
Also, $\Delta W=P\Delta V=nR\Delta T$,
Heat absorbed by the gas to increase its temperature is
$\Delta U=n{C}_V\Delta T$
Hence, fraction of heat used to raise the temperature is given by
$\frac{\Delta U}{\Delta Q}=\frac{\text{Increase in internal energy}}{\text{Total heat aborbed}}=\frac{n{C}_V\Delta T}{n{C}_P\Delta T}$
$\Rightarrow \frac{\Delta U}{\Delta Q}=\frac{C_V}{C_P}=\frac{1}{\gamma }$
Hence, option (b) is the correct answer.