To determine the specific heat of the metal, we need to analyze the situation step by step, using the information provided about the specific gravity of ice, the mass of the metal, and the changes in volume when the metal is immersed in the ice-water mixture.
Understanding the Scenario
We have a metal with a mass of 10 grams at a temperature of 100°C, which is immersed in a mixture of ice and water. The specific gravity of ice is 0.92, which means that ice is less dense than water. When the metal is added, the volume of the mixture decreases by 0.1 cm³. This volume change indicates that the metal is displacing some of the water and ice, leading us to calculate the specific heat of the metal.
Key Concepts
- Specific Gravity: The ratio of the density of a substance to the density of a reference substance (usually water).
- Latent Heat of Fusion: The amount of heat required to convert a unit mass of a solid into a liquid without a change in temperature.
- Heat Transfer: When the metal is immersed, it will lose heat to the ice-water mixture until thermal equilibrium is reached.
Calculating the Volume of Ice Displaced
Since the volume of the mixture decreases by 0.1 cm³, this volume corresponds to the volume of the metal that has submerged in the mixture. To find the mass of ice displaced, we can use the specific gravity of ice:
The density of ice can be calculated as follows:
Density of Ice = Specific Gravity of Ice × Density of Water
Given that the density of water at 0°C is 1 g/cm³, we have:
Density of Ice = 0.92 g/cm³ × 1 g/cm³ = 0.92 g/cm³
Now, we can find the mass of the ice displaced:
Mass of Ice Displaced = Volume Displaced × Density of Ice
Mass of Ice Displaced = 0.1 cm³ × 0.92 g/cm³ = 0.092 g
Heat Transfer Calculations
When the metal is immersed in the ice-water mixture, it loses heat, which is absorbed by the ice to melt it. The heat lost by the metal can be expressed as:
Heat Lost by Metal = Mass of Metal × Specific Heat of Metal × Change in Temperature
Since the metal cools from 100°C to 0°C (the temperature of the ice-water mixture), the change in temperature is:
Change in Temperature = 100°C - 0°C = 100°C
Thus, the heat lost by the metal becomes:
Heat Lost = 10 g × c × 100°C
where c is the specific heat of the metal.
Heat Gained by Ice
The heat gained by the ice can be calculated using the latent heat of fusion:
Heat Gained by Ice = Mass of Ice Displaced × Latent Heat of Fusion
Heat Gained = 0.092 g × 80 cal/g = 7.36 cal
Setting Up the Equation
At thermal equilibrium, the heat lost by the metal equals the heat gained by the ice:
10 g × c × 100°C = 7.36 cal
Solving for Specific Heat
Now, we can solve for c:
c = 7.36 cal / (10 g × 100°C)
c = 7.36 cal / 1000 g°C
c = 0.00736 cal/g°C
Final Thoughts
The specific heat of the metal is approximately 0.00736 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 these principles helps in various applications, from material science to thermodynamics.
