Question

When a gas is compressed in an insulated vessel

• A. its internal energy decreases
• B. its temperature decreases
• C. both its pressure and volume increase
• D. both its temperature and volume increase
• E. both its temperature and internal energy increase

Hint:

In adiabatic compression, work done on the gas increases its internal energy, leading to a rise in temperature.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
The problem describes an adiabatic process. An "insulated vessel" means there is no heat exchange between the gas and its surroundings (\(\Delta Q = 0\)). The gas is being "compressed," which means work is being done on the gas. We need to determine the effect on the gas's internal energy and temperature.
Step 2: Key Formula or Approach:
The First Law of Thermodynamics relates the change in internal energy (\(\Delta U\)) to the heat added to the system (\(\Delta Q\)) and the work done by the system (\(\Delta W\)):
\[ \Delta U = \Delta Q - \Delta W \] For an ideal gas, the internal energy is directly proportional to its absolute temperature (\(U \propto T\)).
Step 3: Detailed Explanation:
1. Identify the process: Since the vessel is insulated, no heat can enter or leave the system. This is an adiabatic process. Therefore, \(\Delta Q = 0\).
2. Analyze the work done: The gas is being compressed. This means the surroundings are doing work {on} the gas. In the convention where \(\Delta W\) is the work done {by} the system, compression implies that the volume decreases, and the work done by the gas is negative (\(\Delta W<0\)).
3. Apply the First Law of Thermodynamics:
\[ \Delta U = \Delta Q - \Delta W \] Substitute the values for our process:
\[ \Delta U = 0 - (\Delta W) \] Since \(\Delta W\) is negative, we have:
\[ \Delta U = -(\text{negative value}) = \text{positive value} \] So, \(\Delta U>0\). This means the internal energy of the gas increases.
4. Relate internal energy and temperature: For an ideal gas, the internal energy is a function of temperature only. An increase in internal energy (\(\Delta U>0\)) directly implies an increase in temperature (\(\Delta T>0\)).
5. Analyze the options:
(A) its internal energy decreases: Incorrect, it increases.
(B) its temperature decreases: Incorrect, it increases.
(C) both its pressure and volume increase: Incorrect, volume decreases during compression.
(D) both its temperature and volume increase: Incorrect, volume decreases.
(E) both its temperature and internal energy increase: Correct, as derived above.
Step 4: Final Answer:
In an adiabatic compression, work is done on the gas, which increases its internal energy. The increase in internal energy leads to a rise in temperature. Therefore, both temperature and internal energy increase. This corresponds to option (E).