One mole of hydrogen, assumed to be ideal, is adiabatically expanded from its initial state `(P_(1), V_(1), T_(1))` to the final state `(P_(2), V_(2), T_(2))`. The decrease in the internal energy of the gas during this process will be given by

Options

A. `C_(V)(T_(1)-T_(2))`
B. `C_(P)(T_(1)-T_(2))`
C. `(C_(P)+C_(V))/(2)(T_(1)-T_(2))`
D. `(C_(P)-C_(V))(T_(1)-T_(2))`

Answer

To solve the problem of finding the decrease in internal energy of one mole of hydrogen during an adiabatic expansion, we can follow these steps:

### Step 1: Understand the Concept of Internal Energy
The internal energy (U) of an ideal gas is related to its temperature. For an ideal gas, the change in internal energy (ΔU) can be expressed as:
\[ \Delta U = n C_v \Delta T \]
where:
- \( n \) = number of moles of the gas
- \( C_v \) = molar heat capacity at constant volume
- \( \Delta T \) = change in temperature

### Step 2: Identify the Variables
In this case, we have:
- \( n = 1 \) mole (as given in the problem)
- \( C_v \) for hydrogen (which can be calculated or looked up)
- \( \Delta T = T_2 - T_1 \) (where \( T_2 \) is the final temperature and \( T_1 \) is the initial temperature)

### Step 3: Substitute the Values
Since we are looking for the decrease in internal energy, we need to consider that the temperature decreases during adiabatic expansion. Therefore, we can express the change in temperature as:
\[ \Delta T = T_2 - T_1 \]
This means:
\[ \Delta U = n C_v (T_2 - T_1) = 1 \cdot C_v (T_2 - T_1) \]

### Step 4: Account for the Decrease in Internal Energy
Since the problem asks for the decrease in internal energy, we can express this as:
\[ \Delta U = C_v (T_2 - T_1) \]
However, since \( T_2 < T_1 \) during the expansion, we can rewrite this as:
\[ \Delta U = C_v (T_2 - T_1) = -C_v (T_1 - T_2) \]

### Step 5: Final Expression
Thus, the decrease in internal energy of the gas during this adiabatic process is given by:
\[ \Delta U = -C_v (T_1 - T_2) \]

### Conclusion
The final expression for the decrease in internal energy is:
\[ \Delta U = C_v (T_1 - T_2) \]