. A closed vessel contains air at a pressure of 23.2 psig and a temperature of 30 degree C. The air is heated at a constant volume to 60 degree C with the atmospheric pressure of 759 mm Hg. what is the final gauge pressure in kPa?

Provide correct solution and answer

To solve this problem, we'll use the ideal gas law and the concept that, for a closed vessel at constant volume, the pressure and temperature relationship is given by Gay-Lussac's Law:

\[
\frac{P_1}{T_1} = \frac{P_2}{T_2}
\]

where:
- \(P_1\) = initial absolute pressure
- \(P_2\) = final absolute pressure
- \(T_1\) = initial absolute temperature in Kelvin
- \(T_2\) = final absolute temperature in Kelvin

### Step-by-Step Solution

1. **Convert Temperatures to Kelvin:**

   The temperatures must be in Kelvin. To convert from Celsius to Kelvin:

   \[
   T_1 = 30^\circ C + 273.15 = 303.15 \, K
   \]

   \[
   T_2 = 60^\circ C + 273.15 = 333.15 \, K
   \]

2. **Convert Initial Gauge Pressure to Absolute Pressure:**

   The given initial gauge pressure is 23.2 psig. Gauge pressure is the pressure relative to atmospheric pressure, so to find the absolute pressure, add the atmospheric pressure.

   The atmospheric pressure is given as 759 mm Hg. We need to convert this to psia (pounds per square inch absolute):

   \[
   \text{1 atm} = 760 \, \text{mm Hg} = 14.696 \, \text{psia}
   \]

   Therefore:

   \[
   \text{Atmospheric pressure} = \frac{759}{760} \times 14.696 \approx 14.676 \, \text{psia}
   \]

   The absolute pressure \(P_1\) is:

   \[
   P_1 = \text{Gauge pressure} + \text{Atmospheric pressure} = 23.2 + 14.676 = 37.876 \, \text{psia}
   \]

3. **Calculate the Final Absolute Pressure Using Gay-Lussac's Law:**

   \[
   \frac{P_1}{T_1} = \frac{P_2}{T_2}
   \]

   Solving for \(P_2\):

   \[
   P_2 = P_1 \times \frac{T_2}{T_1} = 37.876 \times \frac{333.15}{303.15}
   \]

   Let's compute \(P_2\).

4. **Convert the Final Absolute Pressure to Gauge Pressure:**

   Once we have the final absolute pressure \(P_2\), we need to convert it back to gauge pressure (psig):

   \[
   \text{Final gauge pressure (psig)} = P_2 - \text{Atmospheric pressure}
   \]

   Finally, we'll convert the gauge pressure from psig to kPa:

   \[
   1 \, \text{psig} = 6.89476 \, \text{kPa}
   \]

Now let's calculate \(P_2\) and then convert to gauge pressure in kPa.The final gauge pressure in the vessel, after heating the air to 60°C, is approximately **185.8 kPa**.

### Summary of Steps:
1. Converted temperatures to Kelvin.
2. Converted the atmospheric pressure from mm Hg to psia.
3. Calculated the initial absolute pressure.
4. Used Gay-Lussac's Law to find the final absolute pressure.
5. Converted the final absolute pressure back to gauge pressure.
6. Converted gauge pressure to kPa.

Would you like further details, or do you have any questions?

### Related Questions:
1. How do you convert gauge pressure to absolute pressure?
2. What are the assumptions made when applying Gay-Lussac's Law?
3. How does changing the volume of a container affect the pressure at a constant temperature?
4. What would happen if the temperature was decreased instead of increased?
5. How do you convert pressure units from psia to bar?

#### Tip:
Always ensure to use absolute temperatures (Kelvin) when applying gas laws.

A closed vessel contains air at a pressure of 23.2 psig and a temperature of 30 degree C. The air is heated at a constant volume to 60 degree C with the atmospheric pressure of 759 mm Hg. What is the final gauge pressure in kPa?

Learn how to calculate the final gauge pressure of air heated in a closed vessel using Gay-Lussac's Law. This thermodynamics problem involves temperature conversion, pressure conversion, and the relationship between pressure and temperature at constant volume. Suitable for high school students (Grades 10-12).

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