Macmillan Learning Consider an ideal gas with an absolute temperature of T₁. To what temperature would the gas need to be heated to double its pressure? Express the answer in terms of T₁. T₂ = Consider an ideal gas with a volume of V₁. To what volume would the gas need to be compressed to double its pressure? Express the answer in terms of V₁. V/₂=
Macmillan Learning Consider an ideal gas with an absolute temperature of T₁. To what temperature would the gas need to be heated to double its pressure? Express the answer in terms of T₁. T₂ = Consider an ideal gas with a volume of V₁. To what volume would the gas need to be compressed to double its pressure? Express the answer in terms of V₁. V/₂=
Chemistry
10th Edition
ISBN:9781305957404
Author:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste
Publisher:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste
Chapter1: Chemical Foundations
Section: Chapter Questions
Problem 1RQ: Define and explain the differences between the following terms. a. law and theory b. theory and...
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How can we solve these problems with temperature?
![**Understanding Gas Laws: Temperature and Volume Changes**
**1. Temperature and Pressure Relationship**
Consider an ideal gas with an absolute temperature of \( T_1 \). To determine to what temperature the gas needs to be heated to double its pressure, we use the ideal gas law, which states that pressure is directly proportional to temperature when the volume is constant.
Given:
\( T_2 = \) (the temperature at which pressure is doubled)
To find the relationship, we use the formula:
\[ P_2 = 2P_1 \]
Using the ideal gas law:
\[ \frac{P_2}{P_1} = \frac{T_2}{T_1} \]
Solve for \( T_2 \):
\[ T_2 = 2T_1 \]
---
**2. Volume and Pressure Relationship**
Consider an ideal gas with a volume of \( V_1 \). To what volume would the gas need to be compressed to double its pressure?
Given:
\( V_2 = \) (the volume required to double the pressure)
We use Boyle’s Law, which states that the pressure of a given mass of gas is inversely proportional to its volume when temperature is constant.
\[ P_1V_1 = P_2V_2 \]
Given \( P_2 = 2P_1 \), solve for \( V_2 \):
\[ V_2 = \frac{V_1}{2} \]
These fundamental gas law principles can help us understand how gases behave under various conditions and are crucial in fields ranging from chemistry to engineering.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F38e21668-df36-4535-9b97-aa4a56ba16f0%2F8d2cee89-5108-4629-8bab-cd72fa52d94e%2Fjyj0nee_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Understanding Gas Laws: Temperature and Volume Changes**
**1. Temperature and Pressure Relationship**
Consider an ideal gas with an absolute temperature of \( T_1 \). To determine to what temperature the gas needs to be heated to double its pressure, we use the ideal gas law, which states that pressure is directly proportional to temperature when the volume is constant.
Given:
\( T_2 = \) (the temperature at which pressure is doubled)
To find the relationship, we use the formula:
\[ P_2 = 2P_1 \]
Using the ideal gas law:
\[ \frac{P_2}{P_1} = \frac{T_2}{T_1} \]
Solve for \( T_2 \):
\[ T_2 = 2T_1 \]
---
**2. Volume and Pressure Relationship**
Consider an ideal gas with a volume of \( V_1 \). To what volume would the gas need to be compressed to double its pressure?
Given:
\( V_2 = \) (the volume required to double the pressure)
We use Boyle’s Law, which states that the pressure of a given mass of gas is inversely proportional to its volume when temperature is constant.
\[ P_1V_1 = P_2V_2 \]
Given \( P_2 = 2P_1 \), solve for \( V_2 \):
\[ V_2 = \frac{V_1}{2} \]
These fundamental gas law principles can help us understand how gases behave under various conditions and are crucial in fields ranging from chemistry to engineering.
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