Two identical systems, with heat capacity at a constant volume that varies as `C_(v) = bT^(3)` (where b is a constant) are thermally isolated. Initially, one system is at a temperature 100 K and the other is at 200K. The systems are then brought to thermal contact and the combined system is allowed reach thermal equilibrium. If `T_(0)^(4)=nxx10^(8)` then what will be the value of n, where `T_(0)` is the final temperature in kelvin?

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Problem
We have two identical systems with heat capacities that vary as \( C_v = bT^3 \). One system is at 100 K and the other at 200 K. When they are brought into thermal contact, they will exchange heat until they reach thermal equilibrium at a final temperature \( T_0 \).

### Step 2: Write the Expression for Internal Energy
The change in internal energy \( dU \) for a system at constant volume is given by:
\[ dU = C_v dT = bT^3 dT \]

### Step 3: Calculate the Change in Internal Energy for Each System
For the first system (initially at 100 K), the change in internal energy as it heats up from 100 K to \( T_0 \) is:
\[ U_1 = \int_{100}^{T_0} bT^3 dT = \frac{b}{4} \left[ T^4 \right]_{100}^{T_0} = \frac{b}{4} (T_0^4 - 100^4) \]

For the second system (initially at 200 K), the change in internal energy as it cools down from 200 K to \( T_0 \) is:
\[ U_2 = \int_{200}^{T_0} bT^3 dT = \frac{b}{4} \left[ T^4 \right]_{T_0}^{200} = \frac{b}{4} (200^4 - T_0^4) \]

### Step 4: Set the Internal Energies Equal
At thermal equilibrium, the total internal energy before contact equals the total internal energy after contact:
\[ U_1 + U_2 = 0 \]
This leads to:
\[ \frac{b}{4} (T_0^4 - 100^4) = \frac{b}{4} (200^4 - T_0^4) \]

### Step 5: Simplify the Equation
We can cancel \( \frac{b}{4} \) from both sides (assuming \( b \neq 0 \)):
\[ T_0^4 - 100^4 = 200^4 - T_0^4 \]
Rearranging gives:
\[ 2T_0^4 = 200^4 + 100^4 \]
\[ T_0^4 = \frac{200^4 + 100^4}{2} \]

### Step 6: Calculate \( 200^4 \) and \( 100^4 \)
Calculating these values:
\[ 200^4 = (2 \times 10^2)^4 = 16 \times 10^8 \]
\[ 100^4 = (10^2)^4 = 10^8 \]
Thus,
\[ T_0^4 = \frac{16 \times 10^8 + 10^8}{2} = \frac{17 \times 10^8}{2} = 8.5 \times 10^8 \]

### Step 7: Identify \( n \)
From the problem statement, we have:
\[ T_0^4 = n \times 10^8 \]
Comparing this with our result:
\[ n = 8.5 \]

### Final Answer
The value of \( n \) is:
\[ \boxed{8.5} \]