Understanding Specific Heat Capacity: A Mathematical Approach

Introduction

Understanding the specific heat capacity of a metal is crucial in various scientific and engineering applications. This article delves into the mathematical process of determining the specific heat capacity of a metal using the principles of thermodynamics. By analyzing the heat transfer between a heated metal and water, we can calculate the specific heat capacity with a step-by-step approach.

Problem Statement

A piece of metal with a mass of 14.9 g is heated to 98.0°C. When this metal is placed in 75.0 g of water at 20.0°C, the temperature of the water rises by 28.5°C. Using the principle of conservation of energy, we aim to find the specific heat capacity of the metal.

Given Data

Mass of the metal, m_m 14.9 g Initial temperature of the metal, T_{m_i} 98.0°C Mass of water, m_w 75.0 g Initial temperature of the water, T_{w_i} 20.0°C Final temperature of the water and the metal, T_f 48.5°C (28.5°C rise in water temperature from T_{w_i}) Specific Heat Capacity of Water, c_w ≈ 4.18 J/g°C

Heat Gained by Water

The heat gained by the water can be calculated using the formula:

Q_w m_w cdot c_w cdot (T_f - T_{w_i})

Substituting in the known values:

Q_w 75.0 text{g} cdot 4.18 text{J/g°C} cdot (48.5 text{°C} - 20.0 text{°C})

Q_w 75.0 text{g} cdot 4.18 text{J/g°C} cdot 28.5 text{°C}

Q_w ≈ 89.0 text{J}

Heat Lost by Metal

The heat lost by the metal is given by:

Q_m m_m cdot c_m cdot (T_{m_i} - T_f)

Setting the heat lost by the metal equal to the heat gained by the water:

m_m cdot c_m cdot (T_{m_i} - T_f) Q_w

Substituting the known values:

14.9 text{g} cdot c_m cdot (98.0 text{°C} - 48.5 text{°C}) 89.0 text{J}

14.9 text{g} cdot c_m cdot 49.5 text{°C} 89.0 text{J}

Solving for c_m:

c_m frac{89.0 text{J}}{14.9 text{g} cdot 49.5 text{°C}}

c_m ≈ 0.120 text{J/g°C}

Conclusion

The specific heat capacity of the metal is approximately 0.120 J/g°C. By applying the principles of conservation of energy and understanding the heat transfer between different substances, we can accurately calculate the specific heat capacity of the metal.

Further Reading

For a deeper understanding of the principles of thermodynamics, I recommend studying the first chapters of your textbook. Focus on control volumes, closed systems, and the energy balance equation. These concepts are fundamental in more advanced applications of thermodynamics in engineering and applied science.