Question

Which one of the following is the correct relation between $C_p$ and $C_V$ for one mole of an ideal gas? (R is molar gas constant)

• A. $C_p = C_V - R$
• B. $C_p = C_V + R$
• C. $C_p = R - C_V$
• D. $C_p = C_V \times R$
• E. $C_p = C_V / R$

Hint:

Always remember $C_p$ is greater than $C_V$ because work is done during constant pressure heating.

Answer 1

The Correct Option is B

The given question asks to determine the correct relation between the molar heat capacities at constant pressure (\(C_p\)) and at constant volume (\(C_V\)) for one mole of an ideal gas, with \(R\) being the molar gas constant.

First, let's understand the context:

• For an ideal gas, the molar heat capacity at constant volume, \(C_V\), is defined as the amount of heat required to raise the temperature of one mole of the gas by one degree Celsius (or Kelvin) at constant volume.
• The molar heat capacity at constant pressure, \(C_p\), is defined similarly, but at constant pressure.
• The relation between \(C_p\) and \(C_V\) comes from the first law of thermodynamics and is derived as follows:

The first law of thermodynamics, in terms of heat capacities for one mole, can be expressed as:

• \( C_p - C_V = \left( \frac{\delta Q}{dT} \right)_p - \left( \frac{\delta Q}{dT} \right)_V \)

For one mole of an ideal gas, the difference between the two capacities is equal to the ideal gas constant \(R\):

\(C_p - C_V = R\)

Thus, we can rearrange this equation to find the relationship:

\(C_p = C_V + R\)

Therefore, the correct option is \(C_p = C_V + R\).

• This option satisfies the known relation derived from the kinetic theory of gases and is based on the assumption that gas behaves ideally.

Let's consider the options given:

1. \(C_p = C_V - R\) - Incorrect, as it is the opposite of the correct relation.
2. \(C_p = C_V + R\) - Correct, matches the derived equation.
3. \(C_p = R - C_V\) - Incorrect, does not represent the correct relationship.
4. \(C_p = C_V \times R\) - Incorrect, as the relation involves addition, not multiplication.
5. \(C_p = C_V / R\) - Incorrect, the equation involves addition rather than division.

Therefore, the correct answer is \(C_p = C_V + R\).