An ideal gas with heat capacity at constant v | vedclass

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(C) The process equation is $p V^{\alpha} = K$ (constant).From the ideal gas law,$pV = nRT$,so $p = \frac{nRT}{V}$.Substituting this into the process

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$C_V + \frac{nR}{1-\alpha}$

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(C) The process equation is $p V^{\alpha} = K$ (constant).From the ideal gas law,$pV = nRT$,so $p = \frac{nRT}{V}$.Substituting this into the process equation: $\left(\frac{nRT}{V}ight) V^{\alpha} = K \Rightarrow T V^{\alpha-1} = 	ext{constant}$.Differentiating with respect to $T$,we relate the change in temperature $\Delta T$ to the change in volume $\Delta V$.The work done in a polytropic process $p V^{\alpha} = K$ is given by $\Delta W = \int p dV = \frac{p_f V_f - p_i V_i}{1-\alpha} = \frac{nR \Delta T}{1-\alpha}$.According to the first law of thermodynamics,$\Delta Q = \Delta U + \Delta W$.We know $\Delta Q = C \Delta T$ and $\Delta U = C_V \Delta T$.Substituting these into the first law equation: $C \Delta T = C_V \Delta T + \frac{nR \Delta T}{1-\alpha}$.Dividing by $\Delta T$,we get the molar heat capacity $C = C_V + \frac{nR}{1-\alpha}$.

An ideal gas with heat capacity at constant volume $C_V$ undergoes a quasistatic process described by $p V^{\alpha} = \text{constant}$ in a $p-V$ diagram,where $\alpha$ is a constant. The heat capacity of the gas during this process is given by

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