Thermal Physics - Internal Energy

In summary: R/(γ-1). Similarly, for cp, we can write it as cp = Rγ/(γ-1). Therefore, the ratio cp/cv can be written as:cp/cv = (Rγ/(γ-1))/(R/(γ-1)) = γThis shows that the ratio cp/cv is independent of temperature and pressure, as it only depends on the value of γ, which is a constant for a given gas.(c) To show that the equation of an adiabatic change has the form p(V-b)^γ = const, we can use the adiabatic equation, which relates pressure (p), volume (V), and temperature (
  • #1
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Homework Statement



A gas obeys the equation p(V-b) = RT and has cv independent of temperature. Show:

(a) the internal energy is a function of temperature only
(b) the ratio cp/cv is independent of temperature and pressure
(c) the equation of an adiabatic change has the form p(V-b)^gamma = const (where gamma = cp/cv )


Homework Equations





The Attempt at a Solution



Not sure where to go for this one...

I know that for an ideal gas U = 3/2 n R T

But what can I use here? Where do i start? ( can i use dU = TdS - pdV? That holds for all gases, right?)
 
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  • #2


Thank you for your question. I will do my best to provide a thorough and informative response.

(a) To show that the internal energy is a function of temperature only, we can use the first law of thermodynamics, which states that the change in internal energy (ΔU) of a system is equal to the heat added to the system (Q) minus the work done by the system (W). In mathematical terms, this can be written as:

ΔU = Q - W

For an ideal gas, the work done can be expressed as:

W = -PΔV

Substituting this into the first law equation, we get:

ΔU = Q + PΔV

Now, for a gas that obeys the equation p(V-b) = RT, we can write the change in volume (ΔV) as:

ΔV = V2 - V1 = (RT2/p2) - (RT1/p1) = R(T2/p2 - T1/p1)

Substituting this into the first law equation, we get:

ΔU = Q + PR(T2/p2 - T1/p1)

Since cv is independent of temperature, we can write it as a constant, cv = R/(γ-1), where γ = cp/cv. Therefore, the first law equation becomes:

ΔU = Q + (γ-1)Q = γQ

Since Q is the heat added to the system, we can write it as Q = nCvΔT, where n is the number of moles of gas and ΔT is the change in temperature. Substituting this into the first law equation, we get:

ΔU = nCvΔT + (γ-1)nCvΔT = nCv(γΔT)

Since Cv is a constant, we can write it outside the parentheses, and we get:

ΔU = nCvΔT(γ)

This shows that the change in internal energy (ΔU) is directly proportional to the change in temperature (ΔT), and therefore, the internal energy (U) is a function of temperature only.

(b) To show that the ratio cp/cv is independent of temperature and pressure, we can use the definition of γ = cp/cv. From part (a), we know that cv is a
 

What is thermal physics?

Thermal physics is a branch of physics that deals with the study of heat and its relationship to other forms of energy. It focuses on understanding the properties and behavior of matter at the microscopic level in relation to thermal energy.

What is internal energy?

Internal energy is the total energy contained within a system, including the kinetic and potential energy of its particles. It is a measure of the microscopic energy of a substance and is affected by factors such as temperature, pressure, and phase changes.

How is internal energy related to temperature?

The internal energy of a system is directly proportional to its temperature. As the temperature of a substance increases, its internal energy also increases because the particles within the substance have higher kinetic energy.

What is the difference between heat and internal energy?

Heat is a form of energy that is transferred from one system to another due to a temperature difference. Internal energy, on the other hand, is the total energy contained within a system. Heat can change the internal energy of a system, but they are not the same thing.

How is internal energy conserved in a closed system?

According to the law of conservation of energy, energy cannot be created or destroyed, only transferred from one form to another. In a closed system, where there is no exchange of matter or energy with the surroundings, the total internal energy remains constant.

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