Differentiate between Cp and Cv.

In thermodynamics, Cp (specific heat at constant pressure) and Cv (specific heat at constant volume) are two important thermodynamic properties that describe the heat capacity of a substance. They represent the amount of heat required to raise the temperature of a given amount of substance by one degree Celsius (or one Kelvin) at constant pressure and constant volume, respectively. The key difference between Cp and Cv lies in the conditions under which the heat is added or removed.

Cp (Specific Heat at Constant Pressure):
Cp is defined as the amount of heat required to raise the temperature of a substance by one degree Celsius (or one Kelvin) at constant pressure. It is generally higher than Cv for most substances because at constant pressure, some of the heat added is used to do work on the surroundings by expansion (e.g., the substance expands against atmospheric pressure).
The mathematical expression for Cp is given by:
Cp = (∆Q / ∆T) at constant pressure

where:
∆Q = Heat added or removed
∆T = Change in temperature

Cp is used in various thermodynamic calculations, such as the calculation of heat transfer in systems with constant pressure, determining the enthalpy change in chemical reactions, and analyzing systems with pressure variations.

Cv (Specific Heat at Constant Volume):
Cv is defined as the amount of heat required to raise the temperature of a substance by one degree Celsius (or one Kelvin) at constant volume. At constant volume, there is no expansion work performed, so all the heat added is used to increase the internal energy of the substance.
The mathematical expression for Cv is given by:
Cv = (∆Q / ∆T) at constant volume

where:
∆Q = Heat added or removed
∆T = Change in temperature

Cv is particularly useful in situations where the volume of the system is constant, such as in closed containers or when the substance is confined within rigid boundaries.

The relationship between Cp and Cv is given by the specific gas constant (R) for a particular substance:
Cp - Cv = R

For ideal gases, the specific gas constant (R) is constant and depends only on the gas itself, not on the conditions. However, for real gases, R may slightly vary with temperature and pressure.

In summary, Cp and Cv are both measures of the heat capacity of a substance, but they represent heat capacities at different conditions: constant pressure (Cp) and constant volume (Cv). The value of Cp is typically higher than Cv due to the additional work done during expansion at constant pressure.