Question

How much heat is required to completely melt 500 g gold earrings, given that their initial temperature is \(25^{\circ} \mathrm{C} ?\) (The melting point of gold is \(1064^{\circ} \mathrm{C}\), its heat of fusion is \(1.05 \times 10^{5} \frac{\mathrm{J}}{\mathrm{kg}}\). and its specific heat is \(126 \frac{\mathrm{J}}{\mathrm{kg} \cdot \mathrm{K}}\) a. \(15 \mathrm{kJ}\) b. \(32 \mathrm{kJ}\) c. \(66 \mathrm{kJ}\) d. \(97 \mathrm{kJ}\)

Short answer

No given answer matches the calculated 118 kJ.

Step-by-step solution

- Calculate the Energy Required to Heat the Gold to Its Melting Point

First, determine how much energy is needed to raise the temperature of the gold from room temperature to its melting point. Use the formula \[ Q = mc\Delta T \] where: - \(m\) is the mass (in kg) - \(c\) is the specific heat capacity - \( \Delta T \) is the change in temperature Given: \( m = 0.5 \) kg (500 g) \( c = 126 \ \frac{J}{kg \cdot K} \) \( \Delta T = 1064^{\circ} \mathrm{C} - 25^{\circ} \mathrm{C} = 1039 K \) Substitute the values: \[ Q = (0.5 \ kg) \times (126 \ \frac{J}{kg \cdot K}) \times (1039 \ K) \] \[ Q = 65517 \ \text{J} \]

- Calculate the Energy Needed for the Phase Change (Melting)

To find the energy required to melt the gold after it reaches its melting point, use the formula \[ Q = mL \] where: - \(m\) is the mass (in kg) - \(L\) is the latent heat of fusion Given: \( m = 0.5 \) kg (500 g) \( L = 1.05 \times 10^5 \ \frac{J}{kg} \) Substitute the values: \[ Q = 0.5 \ kg \times 1.05 \times 10^5 \ \frac{J}{kg} \] \[ Q = 52500 \ \text{J} \]

- Sum the Energies to Find the Total Heat Required

Add the energy needed to heat the gold to its melting point (from Step 1) with the energy needed to melt the gold (from Step 2): \[ Q_{\text{total}} = 65517 \ \text{J} + 52500 \ \text{J} \] \[ Q_{\text{total}} = 118017 \ \text{J} \] Convert the energy to kJ: \[\frac{118017 \ \text{J}}{1000} = 118 \ \text{kJ} \]

- Compare with Given Options

Compare the calculated value to the given answer choices: a. 15 kJ b. 32 kJ c. 66 kJ d. 97 kJ The calculated value of 118 kJ does not match any of the provided options, indicating either a recheck is needed or a possible error in the given options.

Key concepts

Specific Heat Capacity

Specific heat capacity is the amount of heat required to raise the temperature of 1 kilogram of a substance by 1 degree Celsius (or 1 Kelvin). It’s a crucial concept in thermodynamics, and it tells us how a substance reacts to heat.
For gold, the specific heat capacity is given as 126 J/(kg·K). Here’s what that means practically:

• If you have 1 kilogram of gold and you want to raise its temperature by 1 K, you need 126 joules of energy.
• In our problem, we need to calculate how much energy is required to heat 0.5 kilograms of gold from 25°C to its melting point, which is 1064°C. This involves a temperature change of 1039 K.
• Using the formula: \[ Q = mc\triangle T \] we substitute in the values to find the heat required: \ Q = (0.5 \text{ kg}) \times (126 \frac{J}{kg \text{K}}) \times (1039 \text{ K}) \. The result is 65517 J.

Latent Heat of Fusion

Latent heat of fusion refers to the amount of heat required to change 1 kilogram of a solid into a liquid without changing its temperature. This is an essential concept when dealing with phase changes.
For gold, the latent heat of fusion is provided as 105,000 J/kg. This implies:

• To melt 1 kilogram of gold, you need 105,000 joules of energy.
• In our exercise, we need to melt 0.5 kilograms of gold. Using the given latent heat of fusion, the calculation is: \[ Q = mL \] Substituting the values, we get: \ Q = (0.5 \text{ kg}) \times (1.05 \times 10^5 \frac{J}{kg}) \. This results in 52500 J.

Phase Change

A phase change is when a substance changes from one state of matter to another, such as from solid to liquid. This process involves significant energy but does not change the temperature of the substance while the phase change occurs.
Here, gold transitions from a solid at its melting point (1064°C) to a liquid. To calculate the total heat required for this process, we combine two forms of energy:

• First, the energy to raise the temperature of the gold to its melting point.
• Second, the energy needed for the actual phase change (melting).

Summing these, we get:
\[ Q_{\text{total}} = Q_{\text{heating}} + Q_{\text{melting}} \] Using our values from the specific heat capacity and the latent heat of fusion: \ Q_{\text{total}} = 65517 \text{ J} + 52500 \text{ J} = 118017 \text{ J} \.
Convert this to kilojoules: \ 118017 \text{ J} / 1000 = 118 \text{ kJ} \. Interestingly, none of the answer choices match this value, suggesting a potential error in the given options or a need to recheck the calculations. Nonetheless, the concept of phase change remains central, as it illustrates how different forms of heat applications can cumulatively affect the state and temperature of a substance.