Temperature Mixing Liquid Solids

Overview

Engineering reference data for Temperature Mixing Liquid Solids in thermodynamics.

Key Formulas

First Law

$\Delta U = Q - W$

Energy is conserved — heat added minus work done.

Ideal Gas Law

$PV = nRT$

Relates pressure, volume, and temperature of an ideal gas.

Heat Transfer

$Q = mc\Delta T$

Sensible heat transfer.

Carnot Efficiency

$\eta = 1 - T_C/T_H$

Maximum efficiency between two temperatures.

Variables

Example: Mixing a Heated Stone in Water

Consider a practical example where a granite stone is heated and mixed with water. The stone has a mass of $m_1 = 1 \, \text{kg}$ and specific heat capacity $cp_1 = 790 \, \text{J/kg°C}$. It is heated to an initial temperature of $t_1 = 100 \, \text{°C}$. The water has a mass of $m_2 = 10 \, \text{kg}$, specific heat capacity $cp_2 = 4186 \, \text{J/kg°C}$, and initial temperature $t_2 = 20 \, \text{°C}$.

Using the formula for final mixed temperature, we substitute the values:

$$t_f = \frac{m_1 cp_1 t_1 + m_2 cp_2 t_2}{m_1 cp_1 + m_2 cp_2}$$ $$t_f = \frac{(1 \, \text{kg}) (790 \, \text{J/kg°C}) (100 \, \text{°C}) + (10 \, \text{kg}) (4186 \, \text{J/kg°C}) (20 \, \text{°C})}{(1 \, \text{kg}) (790 \, \text{J/kg°C}) + (10 \, \text{kg}) (4186 \, \text{J/kg°C})}$$

First, calculate the numerator:
$1 \times 790 \times 100 = 79,000 \, \text{J}$
$10 \times 4186 \times 20 = 837,200 \, \text{J}$
Total numerator: $79,000 + 837,200 = 916,200 \, \text{J}$.

Next, calculate the denominator:
$1 \times 790 = 790 \, \text{J/°C}$
$10 \times 4186 = 41,860 \, \text{J/°C}$
Total denominator: $790 + 41,860 = 42,650 \, \text{J/°C}$.

Thus, the final mixed temperature is:

$$t_f = \frac{916,200 \, \text{J}}{42,650 \, \text{J/°C}} \approx 21.5 \, \text{°C}$$

This example demonstrates the application of the mixing formula to find the equilibrium temperature when solids and liquids interact thermally.

Calculator and Spreadsheet Tools

For automated calculations, an online mixed temperature calculator based on the core formula is available. Additionally, a Google Docs spreadsheet can be used to compute final temperatures when mixing multiple liquids or solids; this spreadsheet can be copied or downloaded for offline use and customized as needed. These tools simplify complex calculations and reduce manual errors.

References

• Original Source