Determine the temperature on the second day, assuming that the pressure and amount of gas in a natural gas storage tank have not changed, where the tank is a cylinder with a moveable top and a fixed radius. The height of the cylinder is 22.6 m at 22 °C, and the next day the height increases to 23.8 m due to a heat wave.

Verified step by step guidance

1. Identify the relationship between the volume of the gas and temperature using Charles's Law, which states that the volume of a gas is directly proportional to its temperature when pressure and the amount of gas are constant.
2. Express Charles's Law mathematically as \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \), where \( V_1 \) and \( V_2 \) are the initial and final volumes, and \( T_1 \) and \( T_2 \) are the initial and final temperatures in Kelvin.
3. Calculate the initial volume \( V_1 \) using the formula for the volume of a cylinder: \( V_1 = \pi r^2 h_1 \), where \( h_1 = 22.6 \) m.
4. Calculate the final volume \( V_2 \) using the formula for the volume of a cylinder: \( V_2 = \pi r^2 h_2 \), where \( h_2 = 23.8 \) m.
5. Convert the initial temperature from Celsius to Kelvin by adding 273.15 to 22 °C, then use Charles's Law to solve for the final temperature \( T_2 \) in Kelvin, and convert it back to Celsius if needed.

Key Concepts

Ideal Gas Law

The Ideal Gas Law relates the pressure, volume, temperature, and amount of gas in a system through the equation PV = nRT. In this scenario, since pressure and the amount of gas remain constant, changes in temperature can be inferred from changes in volume, which is represented by the height of the gas in the cylinder.

Charles's Law

Charles's Law states that the volume of a gas is directly proportional to its temperature when pressure is held constant. This principle is crucial for understanding how the increase in height of the gas in the cylinder corresponds to an increase in temperature, allowing us to calculate the new temperature based on the change in volume.

Temperature Conversion

Temperature in gas law calculations is typically measured in Kelvin. To convert Celsius to Kelvin, one must add 273.15 to the Celsius temperature. This conversion is essential for accurately applying gas laws, as they require absolute temperature values to ensure correct proportional relationships.