The cylinder in FIGURE CP18.73 has a moveable piston attached to a spring. The cylinder's cross-section area is 10 cm², it contains 0.0040 mol of gas, and the spring constant is 1500 N/m. At 20°C the spring is neither compressed nor stretched. How far is the spring compressed if the gas temperature is raised to 100°C?

Verified step by step guidance

1. Convert the given temperatures from Celsius to Kelvin using the formula: \( T(K) = T(°C) + 273.15 \). For 20°C, \( T_1 = 293.15 \ \text{K} \), and for 100°C, \( T_2 = 373.15 \ \text{K} \).
2. Use the ideal gas law \( PV = nRT \) to calculate the initial pressure \( P_1 \) of the gas. Rearrange the formula to \( P_1 = \frac{nRT_1}{V} \), where \( n = 0.0040 \ \text{mol} \), \( R = 8.314 \ \text{J/(mol·K)} \), and \( V \) is the volume of the cylinder. Note that \( V = A \cdot x \), where \( A \) is the cross-sectional area and \( x \) is the displacement of the piston.
3. When the temperature is raised to 100°C, the gas expands, compressing the spring. The new pressure \( P_2 \) can be calculated using \( P_2 = \frac{nRT_2}{V_2} \), where \( V_2 = A \cdot (x + \Delta x) \). Here, \( \Delta x \) is the additional compression of the spring due to the temperature increase.
4. Apply Hooke's law for the spring: \( F = k \Delta x \), where \( F \) is the force exerted by the gas on the piston, \( k = 1500 \ \text{N/m} \) is the spring constant, and \( \Delta x \) is the spring compression. The force exerted by the gas is related to the pressure by \( F = P_2 \cdot A \). Combine this with Hooke's law to find \( \Delta x \): \( P_2 \cdot A = k \Delta x \).
5. Solve the system of equations to find \( \Delta x \). Substitute \( P_2 \) from the ideal gas law into the equation \( P_2 \cdot A = k \Delta x \), and use the relationship between \( V_2 \) and \( \Delta x \) to express everything in terms of \( \Delta x \). Simplify and solve for \( \Delta x \).

Key Concepts

Ideal Gas Law

The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of a gas through the equation PV = nRT. This law is essential for understanding how changes in temperature affect the pressure and volume of the gas in the cylinder, particularly when the gas is heated from 20°C to 100°C.

Spring Force and Hooke's Law

Hooke's Law states that the force exerted by a spring is proportional to its displacement from the equilibrium position, expressed as F = -kx, where k is the spring constant and x is the displacement. This concept is crucial for determining how much the spring compresses in response to the pressure exerted by the gas as its temperature increases.

Thermal Expansion and Gas Behavior

As the temperature of a gas increases, its molecules gain kinetic energy, leading to increased pressure if the volume is constant. Understanding this behavior is vital for predicting how the gas in the cylinder will exert force on the piston and compress the spring when the temperature rises from 20°C to 100°C.