Question
Establish relation between two specific heats of a gas. Which is greater and why?
✓
Answer
Relation between $C_p$ and $C_v$. Suppose one mole of a gas is heated so that its temperature rises by $d T$. Heat supplied $=1 \times C_p \times d T=C_v d T \ldots$...(i) Since the volume is constant, the gas will not perform external work in accordance with the first law of thermodynamics and the heat supplied will be just equal to the increase in the internal energy of the gas. $\mathrm{dU}=\mathrm{C}_{\mathrm{v}} \mathrm{dT}$...(ii) Let the gas be heated at constant pressure to again increase its temperature by dT , and dQ be the amount of heat supplied, therefore, $\therefore \mathrm{dQ}=1 \times \mathrm{C}_p \times d T=C_p d T$...(iii) The heat supplied at a constant pressure increases the temperature by dT hence, increases its internal energy by dU as well as enables the gas to perform work $d W . d W=P V$...(iv) From the first law of thermodynamics, we have, $d Q=d U+d W$ Substituting the values, we get, $C_p d T=C_v d T+$ PDV But PV $=$ RT (For one mole of the gas) $O r$ PdV $=$ RdT $\therefore C_p d T=C_v d T+R d T$ or $C_p-C_v=R \ldots$ (v) This is the relation between two principal specific heats of the gas when $C_p, C_v$ and $R$ are measured in the units of either heat or of work. $C_p>C_v$ because a part of the energy supplied in the adiabatic process goes to increase the volume of the gas and the remaining increases the temperature.
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