Textbook Question
A gas cylinder holds 0.10 mol of O₂ at 150°C and a pressure of 3.0 atm. The gas expands adiabatically until the volume is doubled. What are the final pressure?
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22. The First Law of Thermodynamics
Heat Equations for Special Processes & Molar Specific Heats
Problem 64d
Textbook Question
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 guidance1
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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