# Relationship Between V and T in Adiabatic Expansion

• Chronum
In summary, the conversation discusses the relationship between the energy and pressure of black body radiation as well as the change in these values during an adiabatic process. The equations E = σVT^4 and p = 1/3σT^4 are introduced and the question of finding the relation between dE and dT is posed. After some attempts using a simplistic method, the conversation notes the need to consider the changing volume as well. Eventually, Eq(2) is used along with some algebra and differential equations to arrive at the solution.
Chronum

## Homework Statement

The energy and pressure of black body radiation depend on T and V as Eq(1) & Eq(2). Suppose that the temperature and volume of a box of radiation change adiabatically. Find the relation between dE and dT in this process. Next, using Eq(1), show that T ∝V^-1/3

## Homework Equations

Eq(1): E = σVT^4;
Eq(2): p = 1/3σT^4;
ΔE = Q - W;

Since Q = 0;
ΔE = -W

## The Attempt at a Solution

To begin with we've (a few people working together) have tried what appears to be an overly simple method.
E = σ V T^4
dE/dT = 4 σ V T^3
dE = 4 σ V T^3 dT
V = dE/(4 σ T^3 dT)

∴ V∝T^-1/3

But this seems overly simplistic, especially since volume is changing too. Any formulae/approaches we're missing?

Do you realize that in your final answer V and T are swapped compared with what was given to be proved?
Certainly the method is not valid.
You do not seem to have used Eq 2 at all. I would think you need to use that and some relationship between V, E and p.

haruspex said:
Do you realize that in your final answer V and T are swapped compared with what was given to be proved?
Certainly the method is not valid.
You do not seem to have used Eq 2 at all. I would think you need to use that and some relationship between V, E and p.
That is correct. I apologize. It was a mistake of plain anticlimactic proportions.

And yes, I did end up using Eq(2), and we got the answer after some rather petty algebra and a step of differential equations. Problem solved.

Chronum said:
That is correct. I apologize. It was a mistake of plain anticlimactic proportions.

And yes, I did end up using Eq(2), and we got the answer after some rather petty algebra and a step of differential equations. Problem solved.
Was that petty or pretty?

## What is the relationship between volume and temperature in adiabatic expansion?

In adiabatic expansion, the volume and temperature of a gas are inversely proportional. This means that as the volume of the gas increases, the temperature decreases, and vice versa. This relationship is described by the adiabatic expansion equation: TVγ-1 = constant, where γ is the heat capacity ratio.

Adiabatic expansion is a process in thermodynamics where a gas expands without exchanging heat with its surroundings. This means that there is no transfer of thermal energy between the gas and its environment, resulting in a change in volume and temperature of the gas.

## How does adiabatic expansion differ from isothermal expansion?

In adiabatic expansion, the gas does not exchange heat with its surroundings, while in isothermal expansion, the gas is kept at a constant temperature. This means that in adiabatic expansion, the temperature of the gas changes, while in isothermal expansion, the temperature remains constant.

## What is the significance of the adiabatic expansion process?

The adiabatic expansion process is important in various fields, including thermodynamics, meteorology, and engineering. It is used in the design of engines, turbines, and compressors, and it also plays a role in weather phenomena, such as thunderstorms and tornadoes. Understanding adiabatic expansion is crucial for studying and predicting the behavior of gases in different systems.

## What factors affect the relationship between volume and temperature in adiabatic expansion?

The relationship between volume and temperature in adiabatic expansion is affected by the heat capacity ratio, the initial conditions of the gas (such as pressure and volume), and the work done on or by the gas during the process. Additionally, the type of gas and the presence of any external forces can also influence this relationship.

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