Question 512: Thermodynamics & KTG

Question

A diatomic gas $(\gamma=1.4)$ does 100 J of work when it is expanded isobarically. Then the heat given to the gas $\_\_\_\_$ J.

Accepted Answer

An isobaric process occurs at constant pressure (P). For isobaric process :
The work done is $\mathrm{W}=\mathrm{P} \Delta \mathrm{V}=\mathrm{nR} \Delta \mathrm{T}$
The change in internal energy is $\Delta \mathrm{U}=\mathrm{n} \mathrm{C}_{\mathrm{v}} \Delta \mathrm{T}$
Heat added to the system is, $\mathrm{Q}=\mathrm{n} \mathrm{C}_{\mathrm{p}} \Delta \mathrm{T}$
The ratio of specific heats is defined as $\gamma=\frac{\mathrm{C}_{\mathrm{p}}}{\mathrm{C}_{\mathrm{V}}}$.
Also, from Mayer's relation $\mathrm{C}_{\mathrm{p}}-\mathrm{C}_{\mathrm{v}}=\mathrm{R}$.
The ratio of Heat to Work is:
$$ \frac{Q}{W}=\frac{n C_p \Delta T}{n R \Delta T}=\frac{C_p}{R} $$
Substituting $\mathrm{R}=\mathrm{C}_{\mathrm{p}}-\mathrm{C}_{\mathrm{v}}$ :
$$ \frac{Q}{W}=\frac{C_p}{C_p-C_v} $$
$$\Rightarrow $$ $$ \frac{\mathrm{Q}}{\mathrm{~W}}=\frac{\frac{\mathrm{C}_{\mathrm{p}}}{\mathrm{C}_{\mathrm{v}}}}{\left(\frac{\mathrm{C}_{\mathrm{p}}}{\mathrm{C}_{\mathrm{v}}}\right)-1}=\frac{\gamma}{\gamma-1} $$
The work done is given as, $\mathrm{W}=100 \mathrm{~J}$ and the ratio of specific heats $\gamma=1.4$ (for a diatomic gas) So,
$$ \frac{\mathrm{Q}}{100}=\frac{1.4}{1.4-1} $$
$$\Rightarrow $$ $\frac{Q}{100}=\frac{1.4}{0.4}$
$$\Rightarrow $$ $\frac{Q}{100}=3.5$
$$\Rightarrow $$ $$ Q=350 \mathrm{~J} $$
Therefore, the heat given to the gas is 350 J .
Hence, the correct option is $\mathbf{3 5 0}$.

Thermodynamics & KTG

A diatomic gas (\gamma=1.4) does 100 J of work when it is expanded isobarically. Then the heat given to the gas \_\_\_\_ J.

Integer / Numeric Answer Type