If a gas is heated at constant pressure, its isothermal compressibility

Options

A. Remains constant
B. Increases linearly with temperature
C. Decreases linearly with temperature
D. Decreases inversely with temperature

Answer

To solve the problem regarding how the isothermal compressibility of a gas varies with temperature when heated at constant pressure, we can follow these steps:

### Step 1: Understand the definition of isothermal compressibility
Isothermal compressibility (k) is defined as:
\[ k = -\frac{1}{V} \left( \frac{dV}{dP} \right)_T \]
where \( V \) is the volume of the gas, \( P \) is the pressure, and \( T \) is the temperature.

### Step 2: Consider the conditions of the problem
In this case, the gas is heated at constant pressure. This means that as the temperature increases, the volume of the gas will also change due to the gas laws (specifically, Charles's Law).

### Step 3: Apply the ideal gas law
For an ideal gas, the relationship between pressure, volume, and temperature is given by:
\[ PV = nRT \]
where \( n \) is the number of moles of gas, \( R \) is the ideal gas constant, and \( T \) is the absolute temperature.

### Step 4: Differentiate the ideal gas law
At constant pressure, we can express the volume \( V \) in terms of temperature \( T \):
\[ V = \frac{nRT}{P} \]
Now, we differentiate \( V \) with respect to \( P \) while keeping \( T \) constant:
\[ \frac{dV}{dP} = \frac{d}{dP} \left( \frac{nRT}{P} \right) \]
Using the quotient rule:
\[ \frac{dV}{dP} = -\frac{nRT}{P^2} \]

### Step 5: Substitute back into the compressibility formula
Now, substituting \( \frac{dV}{dP} \) back into the equation for isothermal compressibility:
\[ k = -\frac{1}{V} \left( -\frac{nRT}{P^2} \right) \]
This simplifies to:
\[ k = \frac{nRT}{PV^2} \]

### Step 6: Analyze how \( k \) varies with temperature
From the equation \( k = \frac{nRT}{PV^2} \), we can see that as the temperature \( T \) increases, the isothermal compressibility \( k \) also increases, assuming \( n \), \( R \), \( P \), and \( V \) are constant.

### Conclusion
Thus, when a gas is heated at constant pressure, its isothermal compressibility increases with temperature.

---