To find the final temperature when mixing two water samples, we can use the principle of conservation of energy. The heat lost by the hotter water will be equal to the heat gained by the cooler water. Here's a step-by-step explanation:
1. Identify the masses and initial temperatures of the water samples:
- Mass of the first sample, [tex]\( m_1 = 50 \)[/tex] grams
- Initial temperature of the first sample, [tex]\( T_{1_{\text{initial}}} = 20^{\circ}C \)[/tex]
- Mass of the second sample, [tex]\( m_2 = 50 \)[/tex] grams
- Initial temperature of the second sample, [tex]\( T_{2_{\text{initial}}} = 40^{\circ}C \)[/tex]
2. Since both water samples have the same specific heat capacity, we can directly use the concept of weighted averages to find the final equilibrium temperature.
3. The final temperature [tex]\( T_f \)[/tex] of the mixture will be a weighted average of the initial temperatures, considering the masses:
[tex]\[
T_f = \frac{m_1 \cdot T_{1_{\text{initial}}} + m_2 \cdot T_{2_{\text{initial}}}}{m_1 + m_2}
\][/tex]
4. Substitute the given values into the formula:
[tex]\[
T_f = \frac{(50 \, \text{g} \cdot 20^{\circ}\text{C}) + (50 \, \text{g} \cdot 40^{\circ}\text{C})}{50 \, \text{g} + 50 \, \text{g}}
\][/tex]
5. Perform the multiplications:
[tex]\[
T_f = \frac{(1000 + 2000)}{100}
\][/tex]
6. Simplify the expression:
[tex]\[
T_f = \frac{3000}{100}
\][/tex]
7. Calculate the final temperature:
[tex]\[
T_f = 30^{\circ}\text{C}
\][/tex]
Therefore, the final temperature of the mixture will be [tex]\( 30^{\circ}C \)[/tex].
