Question

Show that the internal energy of the air (treated as an ideal gas) contained in a room remains constant as the temperature changes between day and night. Assume that the atmospheric pressure around remains constant and the air in the room maintains this pressure by communicating with the surrounding through the windows, doors, etc.

Answer 1

Internal energy of the air (treated as an ideal gas) contained in a room remains constant as the temperature changes between day and night. This can be shown by-

Assumptions-

• Atmospheric pressure is constant
• Air pressure is maintained

At any temperature internal energy can be given by-

U = nCvT where n is the number of moles and T is the temperature.

We know that air is mostly made up of diatomic molecules.

So, Cv for diatomic molecules = 5R/2

By substituting the values we get

U = 5nRT/2

For an ideal gas we know that

PV = nRT

So, U = 5/2 PV

P and V is constant as given in the question.

So U is also constant

Answer 2

When the temperature changes between day and night, it is proved that the internal energy of the air contained in a room remains constant.

Explanation:

Given Data

To show that - the internal energy of the air contained in a room remains constant as the temperature changes between day and night

At a given temperature, we know internal energy

$U=n C_{y} T$

Home air is basically diatomic molecules, so

$c_{v}=\frac{5}{2} R$

$U=\frac{5 n}{2} R T$

We know that  equation of state

nRT = PV

Where  

P is the pressure  

n is number of moles

R is the gas constant  

T is temperature in Kelvin  

V is volume  

$U=\frac{5}{2}(n R T)$

$U=\frac{5}{2} P V$

Now pressure P is also constant room V = constant

So, U = constant

Therefore the internal energy of the air contained in a room remains constant between day and night where assume that atmospheric pressure remains constant.