Let's address each part of the question one by one.
### Part 1: Doubling the Pressure by Changing Temperature
Consider an ideal gas with an initial absolute temperature [tex]\( T_1 \)[/tex]. According to the Ideal Gas Law, which states [tex]\( PV = nRT \)[/tex]:
1. Initial scenario:
- Pressure [tex]\( P_1 \)[/tex]
- Volume [tex]\( V \)[/tex]
- Temperature [tex]\( T_1 \)[/tex]
So, we have:
[tex]\[
P_1 V = nRT_1
\][/tex]
2. When we want to double the pressure:
- New Pressure [tex]\( 2P_1 \)[/tex]
- Volume [tex]\( V \)[/tex] (unchanged)
- New Temperature [tex]\( T_2 \)[/tex]
According to the Ideal Gas Law:
[tex]\[
2P_1 V = nRT_2
\][/tex]
3. Relating the two equations:
[tex]\[
\frac{2P_1 V}{P_1 V} = \frac{nRT_2}{nRT_1}
\][/tex]
Simplifying, we get:
[tex]\[
2 = \frac{T_2}{T_1}
\][/tex]
Therefore:
[tex]\[
T_2 = 2T_1
\][/tex]
### Part 2: Doubling the Pressure by Changing Volume
Consider an ideal gas with an initial volume [tex]\( V_1 \)[/tex]. According to the Ideal Gas Law, which states [tex]\( PV = nRT \)[/tex]:
1. Initial scenario:
- Pressure [tex]\( P_1 \)[/tex]
- Volume [tex]\( V_1 \)[/tex]
- Temperature [tex]\( T \)[/tex] (unchanged)
So, we have:
[tex]\[
P_1 V_1 = nRT
\][/tex]
2. When we want to double the pressure:
- New Pressure [tex]\( 2P_1 \)[/tex]
- New Volume [tex]\( V_2 \)[/tex]
- Temperature [tex]\( T \)[/tex] (unchanged)
According to the Ideal Gas Law:
[tex]\[
2P_1 V_2 = nRT
\][/tex]
3. Relating the two equations:
[tex]\[
\frac{2P_1 V_2}{P_1 V_1} = \frac{nRT}{nRT}
\][/tex]
Simplifying, we get:
[tex]\[
2V_2 = V_1
\][/tex]
Therefore:
[tex]\[
V_2 = \frac{V_1}{2}
\][/tex]
### Final Answers
Based on the above steps:
[tex]\[
T_2 = 2T_1
\][/tex]
[tex]\[
V_2 = \frac{V_1}{2}
\][/tex]
