In which of these diagrams, the density of an ideal gas remains constant?

Options

A.
B.
C.
D.

Explanation

To determine in which diagram the density of an ideal gas remains constant, we can analyze the relationships between pressure (P), volume (V), temperature (T), and density (ρ) using the ideal gas law and the derived relationships.

### Step-by-Step Solution:

1. **Understanding the Ideal Gas Law**:
The ideal gas law is given by the equation:
\[ PV = nRT \]
where \(n\) is the number of moles, \(R\) is the universal gas constant, and \(T\) is the temperature in Kelvin.

2. **Expressing Density**:
The density (ρ) of the gas can be expressed in terms of mass (m) and volume (V):
\[ \rho = \frac{m}{V} \]
From the ideal gas law, we can express the number of moles \(n\) as:
\[ n = \frac{m}{M} \]
where \(M\) is the molar mass. Substituting this into the ideal gas law gives:
\[ PV = \frac{m}{M}RT \]
Rearranging this, we can express density as:
\[ \rho = \frac{PM}{RT} \]

3. **Analyzing Each Diagram**:
- **Diagram 1**: Constant temperature (T) with increasing pressure (P).
- Since \(T\) is constant, as \(P\) increases, \(\rho\) will also increase. Thus, density is not constant.
- **Diagram 2**: Constant pressure (P) with increasing temperature (T).
- As \(T\) increases, \(\rho\) will decrease. Thus, density is not constant.
- **Diagram 3**: Both pressure (P) and temperature (T) are increasing.
- Here, the relationship is more complex. However, if both P and T increase proportionally, the density can vary. Thus, density is not constant.
- **Diagram 4**: Pressure (P) is directly proportional to temperature (T) with no intercept.
- In this case, if \(P\) increases, \(T\) also increases proportionally, keeping the ratio \(\frac{P}{T}\) constant. Therefore, the density \(\rho\) remains constant.

4. **Conclusion**:
The density of the ideal gas remains constant in **Diagram 4**.

### Final Answer:
The density of the ideal gas remains constant in **Diagram 4**.