A mass `m_1` of a substance of specific heat capacity `c_1` at temperature `t_1` is mixed with a mass `m_2` of other substance of specific heat capacity `c_2` at a lower temperature `t_2`. Deduce the expression for the temperature t of the mixture. State the assumption made, if any.

Explanation

To derive the expression for the temperature \( t \) of the mixture when two substances are mixed, we can follow these steps:

### Step 1: Understand the Problem
We have two masses \( m_1 \) and \( m_2 \) of substances with specific heat capacities \( c_1 \) and \( c_2 \), respectively. The first substance is at a higher temperature \( t_1 \) and the second at a lower temperature \( t_2 \). We need to find the final temperature \( t \) of the mixture.

### Step 2: Apply the Principle of Conservation of Energy
In an isolated system, the total heat lost by the hotter substance must equal the total heat gained by the cooler substance. Therefore, the heat lost by the first substance can be expressed as:
\[
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