A 24-cm-diameter vertical cylinder is sealed at the top by a frictionless 20 kg piston. The piston is 84 cm above the bottom when the gas temperature is 303°C. The air above the piston is at 1.00 atm pressure. What will the height of the piston be if the temperature is lowered to 15°C?

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Convert all given temperatures to Kelvin by using the formula: \( T(K) = T(°C) + 273.15 \). For the initial temperature, \( T_1 = 303 + 273.15 \), and for the final temperature, \( T_2 = 15 + 273.15 \).

Use the ideal gas law in the form \( P V = n R T \). Since the pressure \( P \), the number of moles \( n \), and the gas constant \( R \) remain constant, the relationship simplifies to \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \).

Calculate the initial volume \( V_1 \) of the gas. The volume of a cylinder is given by \( V = \pi r^2 h \), where \( r \) is the radius and \( h \) is the height. Here, \( r = \frac{24}{2} \) cm and \( h_1 = 84 \) cm. Substitute these values into the formula to find \( V_1 \).

Rearrange the simplified gas law equation to solve for the final height \( h_2 \): \( h_2 = h_1 \cdot \frac{T_2}{T_1} \). Substitute the values of \( h_1 \), \( T_1 \), and \( T_2 \) into this equation.

Perform the calculation to determine \( h_2 \), which represents the new height of the piston when the temperature is lowered to 15°C.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Ideal Gas Law

The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of an ideal gas through the equation PV = nRT. In this scenario, the pressure and temperature changes will affect the volume of the gas above the piston, allowing us to determine the new height of the piston when the temperature is lowered.

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 height of the piston changes as the temperature decreases, as it allows us to predict that the volume will decrease with a drop in temperature.

Pressure-Volume Relationship

The pressure-volume relationship in gases indicates that for a given amount of gas at constant temperature, the product of pressure and volume remains constant (Boyle's Law). In this case, as the temperature decreases, the pressure above the piston remains constant, which influences the height of the piston and the volume of the gas.