Question

Obtain the relationship between \(c_{p}\) and \(c_{v}\) for a gas that obeys the equation of state \(p(v-b)=R T\).

Short answer

\( c_p = c_v + R \)

Step-by-step solution

Identify the equation of state

The question provides an equation of state for the gas: \[ p(v - b) = RT \]. Recognize that this is a modified form of the ideal gas law that accounts for the volume excluded by the gas molecules (represented by parameter b).

Define specific heat capacities

Specific heat capacities at constant volume and constant pressure are defined as follows: \[ c_v = \frac{dq}{dT} \bigg|_v \] and \[ c_p = \frac{dq}{dT} \bigg|_p \], where \( q \) is the heat added to the system.

First Law of Thermodynamics

According to the first law of thermodynamics, \[ dq = du + pdv \]. For constant volume, \( dq_v = du \) and for constant pressure, use the equation of state to express \( dv \) in terms of \( dT \).

Internal Energy Change

For an ideal gas, the change in internal energy at constant volume is given by: \[ du = c_v dT \]. Using the modified equation of state, the expression becomes: \[ du = c_v dT \] since \( \frac{\bar{dp}}{dT} = \text{constant} \).

Constant Pressure Heat Capacity

To find the heat added at constant pressure, we use: \[ dq_p = du + pdv \]. From the equation of state, \[ pdv = R dT \] (since \( p = \frac{RT}{v - b} \)). Then, substituting gives: \[ dq_p = c_v dT + R dT \], thus \[ c_p = c_v + R \].

Relationship between cp and cv

We arrive at the final relationship given by: \[ c_p = c_v + R \]. This shows that the specific heat at constant pressure is greater than at constant volume by an amount equal to the gas constant \( R \).

Key concepts

equation of state

The equation of state is a crucial concept in thermodynamics. It describes the relationship between state variables like pressure (p), volume (v), and temperature (T) of a gas. In this exercise, the modified form of the ideal gas law given is \[ p(v - b) = RT \] This equation accounts for the volume occupied by gas molecules, represented by parameter b. In simpler terms, b adjusts the ideal gas law to be more realistic for real gases. This is why it is crucial to understand this modified equation when calculating the relationship between specific heats for such gases.

specific heat capacities

Specific heat capacities tell us how much heat is required to change the temperature of a substance. They are crucial for understanding how a gas behaves under certain conditions. We have two specific heat capacities to consider:

• Specific heat at constant volume (\( c_v \)): The amount of heat required to raise the temperature of a unit mass of gas by one degree at constant volume. Mathematically, it is defined as: \[ c_v = \frac{dq}{dT} \bigg|_v \]
• Specific heat at constant pressure (\( c_p \)): The amount of heat required to raise the temperature of a unit mass of gas by one degree at constant pressure. It is defined as: \[ c_p = \frac{dq}{dT} \bigg|_p \]

Learning the difference between these two can help in understanding the specific conditions under which a gas will change.

First Law of Thermodynamics

The First Law of Thermodynamics is fundamental in physics. It states that energy cannot be created or destroyed, only transformed. The mathematical form is: \[ dq = du + pdv \] This law essentially means the heat added to a system (dq) is used to change the internal energy (du) and to do work (pdv). When the volume is constant, the change in internal energy directly equals the heat added \( dq_v = du \). When pressure is constant, gas work involves the volume change.

internal energy change

Internal energy change is another vital concept tied to thermodynamics. For an ideal gas, the change in internal energy (du) at constant volume can be given by: \[ du = c_v dT \] This relates directly to specific heat capacities, showing how internal energy depends on temperature change. For our modified gas equation, as the volume excludes some space because of molecule size (b), this doesn't affect how we express internal energy change at the fundamental level \( du = c_v dT \).

heat capacity

Heat capacity is all about how a substance absorbs heat. For gases, it's essential to understand that heat capacity at constant pressure (\( c_p \)) is different from that at constant volume (\( c_v \)). We've found, from calculations and knowing specific heats: \[ c_p = c_v + R \] Here, R is the universal gas constant. This relationship indicates that heat capacity at constant pressure is always greater by R compared to constant volume. This inclusion of R accounts for the work done by gas expansion.