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.

Explanation

To 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, we can follow these steps:

### Step 1: Understand the relationship between internal energy, temperature, and pressure
For an ideal gas, the internal energy \( U \) is a function of temperature and the number of moles of gas. The formula for internal energy is given by:
\[ U = n C_v T \]
where:
- \( n \) = number of moles of gas,
- \( C_v \) = specific heat at constant volume,
- \( T \) = absolute temperature.

### Step 2: Identify the specific heat for air
Air is primarily composed of diatomic gases (like \( N_2 \) and \( O_2 \)). For diatomic gases, the specific heat at constant volume \( C_v \) is:
\[ C_v = \frac{5}{2} R \]
where \( R \) is the universal gas constant.

### Step 3: Use the ideal gas law
According to the ideal gas law:
\[ PV = nRT \]
where:
- \( P \) = pressure,
- \( V \) = volume,
- \( n \) = number of moles,
- \( R \) = universal gas constant,
- \( T \) = absolute temperature.

### Step 4: Substitute \( nRT \) in the internal energy equation
From the ideal gas law, we can express \( nRT \) as:
\[ nRT = PV \]
Substituting this into the internal energy equation gives:
\[ U = n C_v T = n \left(\frac{5}{2} R\right) T = \frac{5}{2} nRT = \frac{5}{2} PV \]

### Step 5: Analyze the conditions of the problem
In this scenario, the problem states that:
- The pressure \( P \) is constant (atmospheric pressure remains constant).
- The volume \( V \) is also constant (the air is contained in a room of fixed volume).

### Step 6: Conclude that internal energy remains constant
Since both \( P \) and \( V \) are constant, we can conclude that:
\[ U = \frac{5}{2} PV \]
is also constant. Therefore, even if the temperature \( T \) changes between day and night, the internal energy \( U \) of the air in the room remains constant.

### Final Conclusion
Thus, we have shown that the internal energy of the air in the room remains constant as the temperature changes, given that the pressure and volume are constant.

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