To determine the specific heat capacity of the metal, we can apply the principle of conservation of energy, which states that the heat lost by the metal will equal the heat gained by the water. Let's break this down step by step.
Understanding the Heat Exchange
When the hot metal is placed in the cooler water, heat will flow from the metal to the water until they reach thermal equilibrium. In this scenario, we have:
- Initial temperature of the metal, Tm = 100°C
- Initial temperature of the water, Tw = 20°C
- Final equilibrium temperature, Tf = 30°C
- Mass ratio of water to metal = 5:1
Setting Up the Equation
Let’s denote the mass of the metal as mm and the mass of the water as mw. Given the ratio of their masses, we can express the mass of water as:
mw = 5 * mm
Now, we can use the formula for heat transfer:
Heat lost by metal = Heat gained by water
This can be expressed mathematically as:
mm * cm * (Tm - Tf) = mw * cw * (Tf - Tw)
Substituting Known Values
We know that:
- cw (specific heat capacity of water) = 1 cal/g°C
- Tm = 100°C
- Tw = 20°C
- Tf = 30°C
Substituting these values into the equation gives:
mm * cm * (100 - 30) = (5 * mm) * 1 * (30 - 20)
Simplifying the Equation
We can simplify this equation:
mm * cm * 70 = 5 * mm * 1 * 10
Notice that mm cancels out from both sides:
cm * 70 = 50
Calculating the Specific Heat Capacity
Now, we can solve for cm:
cm = 50 / 70
cm = 0.7143 cal/g°C
Final Thoughts
The specific heat capacity of the metal is approximately 0.714 cal/g°C. This value indicates how much heat energy is required to raise the temperature of one gram of the metal by one degree Celsius. Understanding this concept is crucial in thermodynamics and helps in various applications, from engineering to environmental science.
