If the gas temperature at the steam point is 373.15 K, what is the limiting value of the ratio of the pressures of a gas at the steam point and at triple point of water when the gas is kept at constant volume?

To determine the limiting value of the ratio of the pressures of a gas at the steam point (373.15 K) and at the triple point of water (273.15 K), we can utilize the ideal gas law, which is expressed as \( PV = nRT \). In this case, since we are keeping the volume constant, we can focus on the relationship between pressure and temperature.

Understanding the Ideal Gas Law

The ideal gas law states that for a given amount of gas at a constant volume, the pressure is directly proportional to the temperature. This can be simplified to:

• Pressure (P) is proportional to Temperature (T) when volume (V) and the number of moles (n) are constant.

Setting Up the Ratios

Let’s denote the pressure of the gas at the steam point as \( P_{steam} \) and at the triple point as \( P_{triple} \). According to the ideal gas law, we can express the pressures as follows:

• At the steam point: \( P_{steam} = k \cdot T_{steam} \)
• At the triple point: \( P_{triple} = k \cdot T_{triple} \)

Here, \( k \) is a constant that includes the number of moles and the volume of the gas.

Calculating the Ratio

Now, we can find the ratio of the pressures:

\[ \frac{P_{steam}}{P_{triple}} = \frac{k \cdot T_{steam}}{k \cdot T_{triple}} = \frac{T_{steam}}{T_{triple}} \]

Substituting the Values

We know the temperatures:

• Steam point: \( T_{steam} = 373.15 \, K \)
• Triple point: \( T_{triple} = 273.15 \, K \)

Now, substituting these values into our ratio gives:

\[ \frac{P_{steam}}{P_{triple}} = \frac{373.15}{273.15} \]

Performing the Calculation

Calculating this ratio:

\[ \frac{373.15}{273.15} \approx 1.366 \]

Final Thoughts

The limiting value of the ratio of the pressures of the gas at the steam point to that at the triple point, while keeping the volume constant, is approximately 1.366. This means that at the steam point, the pressure of the gas is about 1.366 times greater than at the triple point of water.