One mole of an ideal gas with heat capacity at constant...

One mole of an ideal gas with heat capacity at constant pressure $C_{P}$ undergoes the process $T=T_{0}+\alpha V$, where $T_{0}$ and $\alpha$ are constants. If its volume increases from $V_{1}$ to $V_{2}$, the amount of heat transferred to the gas is (A) $C_{P} R T_{0} \ln \left(\frac{V_{2}}{V_{1}}\right)$ (B) $\alpha C_{P} \frac{\left(V_{2}-V_{1}\right)}{R T_{0}} \ln \left(\frac{V_{2}}{V_{1}}\right)$ (C) $\alpha C_{P}\left(V_{2}-V_{1}\right)+R T_{0} \ln \left(\frac{V_{2}}{V_{1}}\right)$ (D) $R T_{0} \ln \left(\frac{V_{2}}{V_{1}}\right)+\alpha C_{P}\left(V_{1}-V_{2}\right)$

One mole of an ideal gas with heat capacity at constant pressure $C_{P}$ undergoes the process $T=T_{0}+\alpha V$, where $T_{0}$ and $\alpha$ are constants. If its volume increases from $V_{1}$ to $V_{2}$, the amount of heat transferred to the gas is
(A) $C_{P} R T_{0} \ln \left(\frac{V_{2}}{V_{1}}\right)$
(B) $\alpha C_{P} \frac{\left(V_{2}-V_{1}\right)}{R T_{0}} \ln \left(\frac{V_{2}}{V_{1}}\right)$
(C) $\alpha C_{P}\left(V_{2}-V_{1}\right)+R T_{0} \ln \left(\frac{V_{2}}{V_{1}}\right)$
(D) $R T_{0} \ln \left(\frac{V_{2}}{V_{1}}\right)+\alpha C_{P}\left(V_{1}-V_{2}\right)$

Key Concepts

Ideal Gas Law

The Ideal Gas Law, expressed as PV = nRT, is a fundamental relationship in thermodynamics that connects the pressure, volume, and temperature of an ideal gas. This law is crucial in analyzing processes involving gases and allows us to relate changes in one variable to changes in others when the number of moles and the gas constant are known.

Heat Capacity at Constant Pressure

Heat capacity at constant pressure (CP) is a thermodynamic property that quantifies the amount of heat required to raise the temperature of a system by one degree while maintaining constant pressure. It plays an essential role in determining the heat transfer during processes where pressure remains fixed, and is often used in combination with temperature changes to find the total heat involved in a process.

First Law of Thermodynamics

The First Law of Thermodynamics is a statement of energy conservation, dictating that the change in a system's internal energy is equal to the heat added to the system minus the work done by the system on its surroundings. This principle is fundamental in analyzing energy exchanges in any thermodynamic process, as it ties together heat transfer, work done, and changes in internal energy.

Integration in Thermodynamic Processes

In thermodynamic processes where state variables such as temperature and volume are interdependent, integration is employed to compute quantities like heat or work over an interval. By integrating differential forms of the heat added or work done, one can accurately determine the cumulative effects over the path of the process, especially when the relationship between variables is non-linear or dependent on each other.

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