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Which of the following conditions is required for a small coefficient of expansion in the case of a solid?
A) ${C_p} - {C_V} = R$
B) ${C_p} - {C_V} = 2R$
C) ${C_P}$ is slightly greater than ${C_V}$
D) ${C_P}$ is slightly less than ${C_V}$

Last updated date: 17th Apr 2024
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Hint: The process of production of specific heat from the generation of work from the expansion is measured in constant pressure and when expansion is stopped and no work is generated the specific heat then produced is measured in constant volume.

Complete step by step answer:
The thermal expansion can be defined as the tendency of the material to change its shape, size, area, volume and density. The coefficient of thermal expansion tells us how the size of the object changes with respect to the temperature change. It can be used to measure the change in fraction in size per degree change in temperature at constant pressure. The most important thermal expansion coefficient is the volumetric thermal expansion coefficient. When the temperature of the substance changes, they expand or contract which occurs in all the directions. When the substances expand at the same rate in every direction then that substance is said to be isotropic. If the material is isotropic, the area and volumetric thermal expansion coefficient are approximately twice and three times larger than linear thermal expansion coefficient.
The expansion generates work as force from the pressure displaces the surrounding fluid when the pressure is made constant and a sample is made to expand. The work generated must come from the heat energy provided. The specific heat produced from this process is said to be measured at constant pressure. It is denoted by ${C_P}$.
When the expansion is stopped and hence no work is generated and heat energy that has gone into it must contribute to internal energy of the sample which includes raising the temperature by an extra amount. The specific heat which is obtained in this process is said to be measured at constant volume. It is denoted by ${C_V}$.
The heat capacity can be calculated by –
$C = \dfrac{{\Delta Q}}{{\Delta T}}$
where, $\Delta Q$ is the change in heat and $\Delta T$ is the change in the temperature.
The heat capacity at constant pressure according to the first law of thermodynamics states that, heat supplied to the system will contribute both work and change in internal energy. This heat capacity is called ${C_P}$.
The heat capacity at constant volume states that no work can be done so, heat supplied would contribute only to change in internal energy. This heat capacity is denoted by ${C_V}$.
For a small coefficient of expansion, the temperature of the system should be increased by one kelvin, so that it gets distributed when increasing the internal energy of the system and the volume of the system. Therefore, the constant pressure and amount of heat should be added to the system. So, for this ${C_P}$ should be slightly greater than ${C_V}$.
Hence, the correct option is (C).

Note: When calculating thermal expansion, it is necessary to consider whether the body is free to expand or is constrained. If the body is free to expand, the expansion or strain resulting from an increase in temperature can be simply calculated by using the applicable coefficient of Thermal Expansion.