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Ch 18: A Macroscopic Description of Matter
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoKnight Calc 5th EditionCh 18: A Macroscopic Description of MatterProblema 57a
Capitolo 18, Problema 57a

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?

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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.

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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.
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Ideal Gases and the Ideal Gas Law

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.
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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.
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Unit Conversions
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