The 50 kg circular piston shown in FIGURE P18.57 floats on 0.12 mol of compressed air. What is the piston height h if the temperature is 30°C?

검증된 단계별 안내

Convert the given temperature from Celsius to Kelvin using the formula: \( T(K) = T(°C) + 273.15 \). This ensures the temperature is in the correct unit for thermodynamic calculations.

Use the ideal gas law \( PV = nRT \) to relate the pressure \( P \), volume \( V \), number of moles \( n \), gas constant \( R \), and temperature \( T \). Here, \( n = 0.12 \) mol, \( R = 8.314 \ \text{J/(mol·K)} \), and \( T \) is the temperature in Kelvin.

Express the volume \( V \) of the gas in terms of the piston height \( h \). The volume of the gas is the cross-sectional area of the piston \( A \) multiplied by the height \( h \): \( V = A \cdot h \).

Determine the pressure \( P \) exerted by the piston on the gas. The pressure is due to the weight of the piston and is given by \( P = \frac{F}{A} \), where \( F = mg \) is the weight of the piston (\( m = 50 \ \text{kg} \) and \( g = 9.8 \ \text{m/s}^2 \)).

Combine the expressions for \( P \) and \( V \) into the ideal gas law \( PV = nRT \). Substitute \( P = \frac{mg}{A} \) and \( V = A \cdot h \) into the equation, then solve for \( h \) in terms of the given quantities.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

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 a gas through the equation PV = nRT. Here, P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is the temperature in Kelvin. This law is essential for determining the behavior of gases under varying conditions, such as the compressed air in the piston.

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at equilibrium due to the force of gravity. It is calculated using the formula P = ρgh, where ρ is the fluid density, g is the acceleration due to gravity, and h is the height of the fluid column. In this context, it helps to understand how the weight of the piston affects the pressure of the air inside.

Temperature Conversion

Temperature conversion is crucial for calculations involving the Ideal Gas Law, as temperature must be expressed in Kelvin. The conversion from Celsius to Kelvin is done by adding 273.15 to the Celsius temperature. In this problem, converting the given temperature of 30°C to Kelvin is necessary to apply the Ideal Gas Law correctly.