Ch 19: Work, Heat, and the First Law of Thermodynamics

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 19: Work, Heat, and the First Law of Thermodynamics

Problem 50b

Chapter 19, Problem 50b

2.0 mol of gas are at 30 °C and a pressure of 1.5 atm. How much work must be done on the gas to compress it to one third of its initial volume at constant pressure?

Verified step by step guidance

Step 1: Start by identifying the formula for work done on a gas at constant pressure. The work done, \( W \), is given by \( W = -P \Delta V \), where \( P \) is the pressure and \( \Delta V \) is the change in volume.

Step 2: Express the change in volume, \( \Delta V \), as \( \Delta V = V_f - V_i \), where \( V_i \) is the initial volume and \( V_f \) is the final volume. Since the gas is compressed to one third of its initial volume, \( V_f = \frac{1}{3} V_i \).

Step 3: Substitute \( V_f = \frac{1}{3} V_i \) into the expression for \( \Delta V \): \( \Delta V = \frac{1}{3} V_i - V_i = -\frac{2}{3} V_i \).

Step 4: Use the ideal gas law, \( PV = nRT \), to calculate the initial volume, \( V_i \). Rearrange the equation to \( V_i = \frac{nRT}{P} \), where \( n = 2.0 \) mol, \( R = 0.0821 \, \text{L·atm/(mol·K)} \), \( T = 30 + 273 = 303 \, \text{K} \), and \( P = 1.5 \, \text{atm} \).

Step 5: Substitute \( P \), \( \Delta V \), and \( V_i \) into the work formula \( W = -P \Delta V \). This will give you the work done on the gas during compression. Remember to keep the units consistent throughout the calculation.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 the behavior of gases under various conditions, including changes in volume and pressure, which are relevant to the question about compressing gas.

Work Done on a Gas

The work done on a gas during compression at constant pressure can be calculated using the formula W = PΔV, where W is work, P is pressure, and ΔV is the change in volume. This concept is crucial for determining how much work is required to compress the gas to one third of its initial volume.

Constant Pressure Process

In a constant pressure process, the pressure of the gas remains unchanged while its volume changes. This condition simplifies calculations and is important for understanding how the gas behaves during compression, as it allows for the direct application of the work formula without needing to account for varying pressure.

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Textbook Question

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Textbook Question

An ideal-gas process is described by p=cV^1/2, where c is a constant. 0.033 mol of gas at an initial temperature of 150°C is compressed, using this process, from 300 cm³ to 200 cm³. How much work is done on the gas?

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