To determine the temperature at which the density of nitrogen gas is maximum, we can use the relationship between density, pressure, and temperature. The density of a gas can be expressed using the ideal gas law and the formula for density.
### Step-by-Step Solution:
1. **Understand the Ideal Gas Law**:
The ideal gas law is given by the equation:
\[
PV = nRT
\]
where \(P\) is pressure, \(V\) is volume, \(n\) is the number of moles, \(R\) is the ideal gas constant, and \(T\) is temperature.
2. **Express Moles in Terms of Mass**:
The number of moles \(n\) can be expressed as:
\[
n = \frac{m}{M}
\]
where \(m\) is the mass of the gas and \(M\) is the molar mass.
3. **Substituting Moles into the Ideal Gas Law**:
Substituting \(n\) into the ideal gas law gives:
\[
PV = \frac{m}{M}RT
\]
4. **Rearranging for Density**:
The density \(\rho\) is defined as:
\[
\rho = \frac{m}{V}
\]
Therefore, we can rearrange the ideal gas equation to express density:
\[
P = \frac{m}{V} \cdot \frac{RT}{M} \implies \rho = \frac{PM}{RT}
\]
5. **Finding the Relationship**:
From the equation \(\rho = \frac{PM}{RT}\), we can see that density \(\rho\) is directly proportional to pressure \(P\) and inversely proportional to temperature \(T\):
\[
\rho \propto \frac{P}{T}
\]
6. **Analyzing Given Conditions**:
We need to evaluate the conditions at:
- STP: \(P = 1 \, \text{atm}, T = 298 \, \text{K}\)
- \(273 \, \text{K}\) and \(280 \, \text{K}\)
- \(546 \, \text{K}\) and \(180 \, \text{K}\)
7. **Calculating Density Ratios**:
For each condition, we can calculate the density ratio:
- At STP: \(\rho \propto \frac{1}{298}\)
- At \(273 \, \text{K}\): \(\rho \propto \frac{1}{273}\)
- At \(280 \, \text{K}\): \(\rho \propto \frac{1}{280}\)
- At \(546 \, \text{K}\): \(\rho \propto \frac{1}{546}\)
8. **Comparing Values**:
The smaller the denominator, the larger the density. Thus:
- \(\frac{1}{273}\) is larger than \(\frac{1}{280}\) and \(\frac{1}{546}\).
- Therefore, the maximum density occurs at \(273 \, \text{K}\).
### Conclusion:
The density of nitrogen gas is maximum at \(273 \, \text{K}\).
