Documentation:CHBE Exam Wiki/5.5 - Cv and Cp Specific Heat Capacity

5.5 – Cv and Cp Specific Heat Capacity

5.5.0 – Learning Objectives

By the end of this section you should be able to:

1. Understand the concept of specific heat capacity.
2. Understand the difference between heat capacity at constant pressure ($C_{p}$) and constant volume ($C_{v}$).
3. Understand the relationship between the ideal gas constant, $C_{v}$, and, $C_{p}$.
4. Relate internal energy $U$ with $C_{v}$.

5.5.1 – Introduction

Specific heat capacity is the measure of heat required to raise the temperature of an object. This notebook will cover heat capacity, specific heat capacity and the variants at constant volume and constant pressure.

5.5.2 – Specific Heat Capacity

Specific heat capacity at constant volume and pressure are regarded as $C_{v}$ and $C_{p}$ respectively. In reality, these values are functions of temperature. This will be covered in a separate notebook. For now, we will assume that they are constants.

If working with an monatomic species such as Helium, The value of $C_{v}$ is ${\frac {3}{2}}R$ and the value of $C_{p}$ is ${\frac {5}{2}}R$. Otherwise, these values will be given to you. From the monatomic relationship, it is observed that the general rule is that $C_{p}=C_{v}+R$.

Often, in adiabatic settings, $\gamma$ is defined as the ratio ${\frac {C_{p}}{C_{v}}}$

In closed systems where internal energy $U$ is used, Specific heat at constant volume is used to proportionally relate temperature to an object. $\Delta U=nC_{v}\Delta T$ or $\Delta U=mC_{v}\Delta T$

In open systems, Heat $Q$ is written as $nC_{p}\Delta T$ or $mC_{p}\Delta T$

5.5.3 – Specific Properties

A specific property is obtained by dividing an extensive property by mass. This turns the property to be intensive, i.e. not reliant on the situation.

For test purposes, when you see a specific property remember to divide by its mass.

For example: 10kg of water has a heat capacity of $4.184{\frac {kJ}{kg\cdot \Delta T}}\cdot 10kg\cdot =41.84{\frac {kJ}{\Delta T}}$

Likewise, 100kg of water has a heat capacity of $4.184{\frac {kJ}{kg\cdot \Delta T}}\cdot 100kg\cdot =418.4{\frac {kJ}{\Delta T}}$

It takes 41kJ of energy to raise the temperature of 10kg of water by 1 degree and 418 kJ of energy to raise the temperature of 100kg of water by 1 degree.

Both 10 and 100kg of water have a specific heat capacity of $4.184{\frac {kJ}{kg\Delta T}}$