# Specific heat capacity at constant pressure

• v_pino
In summary, the conversation discusses the difference between specific heat capacity at constant pressure and at constant volume. The formula for specific heat capacity at constant pressure takes into account the work done in expanding the gas, while the formula for specific heat capacity at constant volume does not.
v_pino

specific heat capacity at constant pressure = dU-W / m. dT

dU = change in internal energy
W = work done
m = mass of gas
dT = change in temperature

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However, shouldn't the right hand side equate to specific heat capacity at constant VOLUME?

By saying that it is specific heat capacity at constant pressure, I thought we have already taken into account the energy used to expand the gas.

Specific heat capacity at constant pressure, by definition, is

$$c_P=\frac{1}{m}\left(\frac{\partial H}{\partial T}\right)_P=\frac{1}{m}\left(\frac{\partial (U+PV)}{\partial T}\right)_P=\frac{1}{m}\left(\frac{\partial U}{\partial T}\right)_P+\frac{P}{m}\left(\frac{\partial V}{\partial T}\right)_P$$

If $c_P$ is constant, we can integrate and get

$$c_P=\frac{\Delta U+W}{m\Delta T}$$

In contrast,

$$c_V=\frac{1}{m}\left(\frac{\partial U}{\partial T}\right)_V$$

and, if $c_V$ is constant,

$$c_V=\frac{\Delta U}{m\Delta T}$$

Your understanding is correct. The specific heat capacity at constant pressure is indeed related to the specific heat capacity at constant volume. However, the difference lies in the fact that at constant pressure, work is being done on the system to maintain the pressure, whereas at constant volume, no work is being done. Therefore, the specific heat capacity at constant pressure takes into account the energy used to expand the gas, whereas the specific heat capacity at constant volume does not. Both values are important in understanding the thermodynamic properties of a system.

## 1. What is specific heat capacity at constant pressure?

Specific heat capacity at constant pressure, also known as the specific heat capacity at constant pressure or simply specific heat, is the amount of heat energy required to raise the temperature of a substance by one degree Celsius at constant pressure.

## 2. How is specific heat capacity at constant pressure measured?

Specific heat capacity at constant pressure is typically measured using a calorimeter, which is a device that can accurately measure heat exchange between a substance and its surroundings.

## 3. What is the difference between specific heat capacity at constant pressure and specific heat capacity at constant volume?

The main difference between specific heat capacity at constant pressure and specific heat capacity at constant volume is the pressure at which the heat exchange is occurring. At constant pressure, the volume of the substance can change, while at constant volume, the pressure remains constant.

## 4. Why is specific heat capacity at constant pressure important in thermodynamics?

Specific heat capacity at constant pressure is important in thermodynamics because it helps to determine the amount of heat needed to change the temperature of a substance during a chemical reaction or physical process. It also plays a key role in calculating the efficiency of heat engines.

## 5. What factors can affect the specific heat capacity at constant pressure of a substance?

The specific heat capacity at constant pressure of a substance can be affected by factors such as the molecular structure of the substance, the temperature at which it is being measured, and any phase changes that may occur during the process.

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