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A monoatomic gas at pressure P1 and volume V1 is compressed adiabatically to 1/8th of it's original volume. What is the final pressure of the gas? a) 64P1 b) P1 c) 16P1 d) 32P1?
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Adiabatic Compression of a Monoatomic Gas
To solve this problem, we will use the adiabatic compression equation for an ideal gas:
𝑃1𝑉1^𝛾 = 𝑃2𝑉2^𝛾
Where:
- 𝑃1 is the initial pressure of the gas
- 𝑉1 is the initial volume of the gas
- 𝑃2 is the final pressure of the gas
- 𝑉2 is the final volume of the gas
- 𝛾 is the adiabatic index, which is equal to 5/3 for a monoatomic gas.
Given:
- Initial pressure (𝑃1) = P1
- Initial volume (𝑉1) = V1
- Final volume (𝑉2) = 1/8 𝑉1
Finding the Final Pressure:
Step 1: Substitute the given values into the adiabatic compression equation:
𝑃1𝑉1^𝛾 = 𝑃2𝑉2^𝛾
P1 * V1^𝛾 = 𝑃2 * (1/8 𝑉1)^𝛾
Step 2: Simplify the equation:
P1 * V1^𝛾 = 𝑃2 * (1/8)^𝛾 * 𝑉1^𝛾
Step 3: Cancel out 𝑉1^𝛾 on both sides:
P1 = 𝑃2 * (1/8)^𝛾
Step 4: Divide both sides of the equation by (1/8)^𝛾:
P1 / (1/8)^𝛾 = 𝑃2
Step 5: Simplify the equation:
P1 * (8/1)^𝛾 = 𝑃2
P1 * 8^𝛾 = 𝑃2
Step 6: Since 8^𝛾 = (2^3)^𝛾 = 2^(3𝛾), we can rewrite the equation as:
P1 * 2^(3𝛾) = 𝑃2
Step 7: Substitute the value of 𝛾 (adiabatic index for a monoatomic gas) as 5/3:
P1 * 2^(3 * 5/3) = 𝑃2
P1 * 2^5 = 𝑃2
Step 8: Simplify the equation:
P1 * 32 = 𝑃2
Final Answer:
Therefore, the final pressure (𝑃2) of the gas after adiabatic compression is 32 times the initial pressure (P1). Thus, the answer is option (d) 32P1.
To solve this problem, we will use the adiabatic compression equation for an ideal gas:
𝑃1𝑉1^𝛾 = 𝑃2𝑉2^𝛾
Where:
- 𝑃1 is the initial pressure of the gas
- 𝑉1 is the initial volume of the gas
- 𝑃2 is the final pressure of the gas
- 𝑉2 is the final volume of the gas
- 𝛾 is the adiabatic index, which is equal to 5/3 for a monoatomic gas.
Given:
- Initial pressure (𝑃1) = P1
- Initial volume (𝑉1) = V1
- Final volume (𝑉2) = 1/8 𝑉1
Finding the Final Pressure:
Step 1: Substitute the given values into the adiabatic compression equation:
𝑃1𝑉1^𝛾 = 𝑃2𝑉2^𝛾
P1 * V1^𝛾 = 𝑃2 * (1/8 𝑉1)^𝛾
Step 2: Simplify the equation:
P1 * V1^𝛾 = 𝑃2 * (1/8)^𝛾 * 𝑉1^𝛾
Step 3: Cancel out 𝑉1^𝛾 on both sides:
P1 = 𝑃2 * (1/8)^𝛾
Step 4: Divide both sides of the equation by (1/8)^𝛾:
P1 / (1/8)^𝛾 = 𝑃2
Step 5: Simplify the equation:
P1 * (8/1)^𝛾 = 𝑃2
P1 * 8^𝛾 = 𝑃2
Step 6: Since 8^𝛾 = (2^3)^𝛾 = 2^(3𝛾), we can rewrite the equation as:
P1 * 2^(3𝛾) = 𝑃2
Step 7: Substitute the value of 𝛾 (adiabatic index for a monoatomic gas) as 5/3:
P1 * 2^(3 * 5/3) = 𝑃2
P1 * 2^5 = 𝑃2
Step 8: Simplify the equation:
P1 * 32 = 𝑃2
Final Answer:
Therefore, the final pressure (𝑃2) of the gas after adiabatic compression is 32 times the initial pressure (P1). Thus, the answer is option (d) 32P1.
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A monoatomic gas at pressure P1 and volume V1 is compressed adiabatically to 1/8th of it's original volume. What is the final pressure of the gas? a) 64P1 b) P1 c) 16P1 d) 32P1?
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A monoatomic gas at pressure P1 and volume V1 is compressed adiabatically to 1/8th of it's original volume. What is the final pressure of the gas? a) 64P1 b) P1 c) 16P1 d) 32P1? for NEET 2024 is part of NEET preparation. The Question and answers have been prepared according to the NEET exam syllabus. Information about A monoatomic gas at pressure P1 and volume V1 is compressed adiabatically to 1/8th of it's original volume. What is the final pressure of the gas? a) 64P1 b) P1 c) 16P1 d) 32P1? covers all topics & solutions for NEET 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A monoatomic gas at pressure P1 and volume V1 is compressed adiabatically to 1/8th of it's original volume. What is the final pressure of the gas? a) 64P1 b) P1 c) 16P1 d) 32P1?.
A monoatomic gas at pressure P1 and volume V1 is compressed adiabatically to 1/8th of it's original volume. What is the final pressure of the gas? a) 64P1 b) P1 c) 16P1 d) 32P1? for NEET 2024 is part of NEET preparation. The Question and answers have been prepared according to the NEET exam syllabus. Information about A monoatomic gas at pressure P1 and volume V1 is compressed adiabatically to 1/8th of it's original volume. What is the final pressure of the gas? a) 64P1 b) P1 c) 16P1 d) 32P1? covers all topics & solutions for NEET 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A monoatomic gas at pressure P1 and volume V1 is compressed adiabatically to 1/8th of it's original volume. What is the final pressure of the gas? a) 64P1 b) P1 c) 16P1 d) 32P1?.
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