a) Define specific heat capacity. [1] b) A piece of metal of mass 50 g is heated to 100°C and then quickly transferred to 100 g of water at 20°C. The final temperature of the mixture is found to be…

a) Definition of Specific Heat Capacity

Specific heat capacity is defined as the amount of heat energy required to raise the temperature of 1 gram of a substance by 1°C (or 1 K). It is a measure of how much a given substance resists changes in temperature when heat is added or removed. The unit of specific heat capacity is J/g°C (joules per gram per degree Celsius).

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b) Calculation of Specific Heat Capacity of the Metal

Given:

• Mass of metal, $m_m=50\text{g}$
• Initial temperature of metal, $T_m=100^{\circ }\text{C}$
• Mass of water, $m_w=100\text{g}$
• Initial temperature of water, $T_w=20^{\circ }\text{C}$
• Final temperature of mixture, $T_f=24^{\circ }\text{C}$
• Specific heat capacity of water, $c_w=4.2{\text{J/g}}^{\circ }\text{C}$
• Specific heat capacity of metal, $c_m=?$

Assumptions:

• No heat is lost to the surroundings.
• The system is isolated (closed system).

Principle:

Heat lost by the metal = Heat gained by the water.

Step 1: Calculate heat lost by the metal

The heat lost by the metal as it cools from $100^{\circ }\text{C}$ to $24^{\circ }\text{C}$ is given by: $$Q_{\text{lost}}=m_m\cdot c_m\cdot (T_m-T_f)$$ $$Q_{\text{lost}}=50\cdot c_m\cdot (100-24)$$ $$Q_{\text{lost}}=50\cdot c_m\cdot 76$$

Step 2: Calculate heat gained by the water

The heat gained by the water as it warms from $20^{\circ }\text{C}$ to $24^{\circ }\text{C}$ is given by: $$Q_{\text{gained}}=m_w\cdot c_w\cdot (T_f-T_w)$$ $$Q_{\text{gained}}=100\cdot 4.2\cdot (24-20)$$ $$Q_{\text{gained}}=100\cdot 4.2\cdot 4$$ $$Q_{\text{gained}}=1680\text{J}$$

Step 3: Equate heat lost and heat gained

Since heat lost by the metal is equal to heat gained by the water: $$50\cdot c_m\cdot 76=1680$$

Step 4: Solve for $c_m$

$$c_m=\frac{1680}{50\cdot 76}$$ $$c_m=\frac{1680}{3800}$$ $$c_m=0.442{\text{J/g}}^{\circ }\text{C}$$

Final Answer:

The specific heat capacity of the metal is 0.442 J/g°C.