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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
All textbooksKnight Calc 5th EditionCh 19: Work, Heat, and the First Law of ThermodynamicsProblem 64d
Chapter 19, Problem 64d

14 g of nitrogen gas at STP are adiabatically compressed to a pressure of 20 atm. What is the compression ratio Vmax/Vmin?

Verified step by step guidance
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Step 1: Understand the problem. The compression ratio Vₘₐₓ/Vₘᵢₙ is the ratio of the initial volume (Vₘₐₓ) to the final volume (Vₘᵢₙ) during an adiabatic compression. To solve this, we use the adiabatic relation for an ideal gas: \( P₁ V₁^γ = P₂ V₂^γ \), where \( γ \) is the adiabatic index (ratio of specific heats, \( C_p/C_v \)).
Step 2: Determine the value of \( γ \) for nitrogen gas. Nitrogen is a diatomic gas, so \( γ \) is typically \( \frac{7}{5} = 1.4 \). This value is derived from the degrees of freedom of diatomic molecules.
Step 3: Rearrange the adiabatic relation to find the compression ratio. Using \( P₁ V₁^γ = P₂ V₂^γ \), divide both sides by \( P₁ \) and take the \( γ \)-th root: \( \frac{V₁}{V₂} = \left( \frac{P₂}{P₁} \right)^{1/γ} \). The compression ratio \( Vₘₐₓ/Vₘᵢₙ \) is equal to \( \left( \frac{P₂}{P₁} \right)^{1/γ} \).
Step 4: Substitute the given pressures into the formula. The initial pressure \( P₁ \) is 1 atm (STP), and the final pressure \( P₂ \) is 20 atm. Plug these values into the formula: \( Vₘₐₓ/Vₘᵢₙ = \left( \frac{20}{1} \right)^{1/1.4} \).
Step 5: Simplify the expression to find the compression ratio. Perform the calculation \( \left( 20 \right)^{1/1.4} \) to determine the numerical value of the compression ratio. This step involves evaluating the exponentiation, but the final numerical result is not calculated here.

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Key Concepts

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

Adiabatic Process

An adiabatic process is one in which no heat is exchanged with the surroundings. In such processes, the internal energy of the system changes due to work done on or by the system. For gases, this means that during compression or expansion, the temperature and pressure change without heat transfer, which is crucial for understanding how the gas behaves under these conditions.
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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. This law is fundamental in determining the state of a gas under various conditions. In the context of the question, it helps to calculate the initial and final volumes of nitrogen gas when subjected to changes in pressure and temperature during the adiabatic compression.
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Compression Ratio

The compression ratio is defined as the ratio of the maximum volume (Vₘₐₓ) to the minimum volume (Vₘᵢₙ) of a gas during compression. It is a critical parameter in thermodynamics and engineering, indicating how much a gas is compressed. In this scenario, calculating the compression ratio helps to understand the extent of volume reduction and the resulting changes in pressure and temperature of the nitrogen gas.
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Related Practice
Textbook Question

Liquid helium, with a boiling point of 4.2 K, is used in ultralow-temperature experiments and also for cooling the superconducting magnets used in MRI imaging in medicine. Storing liquid helium so far below room temperature is a challenge because even a small 'heat leak' will boil the helium away. A standard helium dewar, shown in FIGURE P19.67, has an inner stainless-steel cylinder filled with liquid helium surrounded by an outer cylindrical shell filled with liquid nitrogen at –196°C. The space between is a vacuum. The small structural supports have very low thermal conductivity, so you can assume that radiation is the only heat transfer between the helium and its surroundings. Suppose the helium cylinder is 16 cm in diameter and 30 cm tall and that all walls have an emissivity of 0.25. The density of liquid helium is 125 kg/m3 and its heat of vaporization is 2.1×104 J/kg. What is the mass of helium in the filled cylinder?

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Textbook Question

FIGURE P19.62 shows a thermodynamic process followed by 120 mg of helium. How much heat energy is transferred to or from the gas during each of the three segments?

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Textbook Question

Two cylinders each contain 0.10 mol of a diatomic gas at 300 K and a pressure of 3.0 atm. Cylinder A expands isothermally and cylinder B expands adiabatically until the pressure of each is 1.0 atm. What are the final temperature and volume of each?

Textbook Question

A monatomic gas is adiabatically compressed to 1/8 of its initial volume. Does each of the following quantities change? If so, does it increase or decrease, and by what factor? If not, why not? The molar specific heat at constant volume.

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Textbook Question

14 g of nitrogen gas at STP are pressurized in an isochoric process to a pressure of 20 atm. What are the final temperature?

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Textbook Question

FIGURE P19.62 shows a thermodynamic process followed by 120 mg of helium. How much work is done on the gas during each of the three segments?

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