One mole of a monatomic ideal gas undergoes an adiabatic expansion in which its volume becomes eight times its initial value. If the initial temperature of the gas is 100 universal gas constant 8.0, the decrease in its internal energy, in , is__________.

Explanation

To solve the problem, we need to determine the decrease in internal energy of one mole of a monatomic ideal gas that undergoes an adiabatic expansion, where its volume increases to eight times its initial value.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Initial temperature, \( T_1 = 100 \, K \)
- Volume change: \( V_2 = 8 V_1 \)
- For a monatomic ideal gas, the heat capacity ratio \( \gamma = \frac{C_p}{C_v} = \frac{5}{3} \).

2. **Use the Adiabatic Condition:**
The adiabatic condition for an ideal gas can be expressed as:
\[ T_1 V_1^{\gamma - 1} = T_2 V_2^{\gamma - 1} \]
Here, \( \gamma - 1 = \frac{5}{3} - 1 = \frac{2}{3} \).

3. **Substituting the Known Values:**
Substitute \( V_2 = 8 V_1 \) into the adiabatic condition:
\[ T_1 V_1^{\frac{2}{3}} = T_2 (8 V_1)^{\frac{2}{3}} \]
Simplifying gives:
\[ T_1 V_1^{\frac{2}{3}} = T_2 (8^{\frac{2}{3}} V_1^{\frac{2}{3}}) \]
\[ T_1 = T_2 \cdot 8^{\frac{2}{3}} \]

4. **Calculate \( 8^{\frac{2}{3}} \):**
\[ 8^{\frac{2}{3}} = (2^3)^{\frac{2}{3}} = 2^2 = 4 \]
Thus, the equation becomes:
\[ T_1 = 4 T_2 \]

5. **Find \( T_2 \):**
Substitute \( T_1 = 100 \, K \):
\[ 100 = 4 T_2 \implies T_2 = \frac{100}{4} = 25 \, K \]

6. **Calculate the Change in Internal Energy:**
The change in internal energy \( \Delta U \) for one mole of a monatomic ideal gas can be calculated using the formula:
\[ \Delta U = \frac{3}{2} n R (T_2 - T_1) \]
Here, \( n = 1 \) mole and \( R = 8 \, J/(mol \cdot K) \):
\[ \Delta U = \frac{3}{2} \cdot 1 \cdot 8 \cdot (25 - 100) \]
\[ \Delta U = \frac{3}{2} \cdot 8 \cdot (-75) \]
\[ \Delta U = \frac{3 \cdot 8 \cdot (-75)}{2} = -900 \, J \]

7. **Final Answer:**
The decrease in internal energy is \( 900 \, J \).