Calculate the minimum amount of energy, in joules, required to completely melt 85.0g of silver initially at `15.0^(@)C`.

Answer

To calculate the minimum amount of energy required to completely melt 85.0 g of silver initially at 15.0 °C, we need to follow these steps:

### Step 1: Convert mass from grams to kilograms
The mass of silver is given as 85.0 g. To convert grams to kilograms, we divide by 1000.

\[ M = \frac{85.0 \, \text{g}}{1000} = 0.085 \, \text{kg} \]

**Hint:** Remember that 1 kg = 1000 g.

### Step 2: Determine the specific heat capacity of silver
The specific heat capacity (c) of silver is given as 236 J/(kg·K).

**Hint:** Specific heat capacity is the amount of heat required to raise the temperature of 1 kg of a substance by 1 K.

### Step 3: Calculate the temperature change (ΔT)
The initial temperature (T_initial) is 15.0 °C, and the melting point of silver (T_melting) is 1235 °C. We need to convert these temperatures to Kelvin for calculations.

\[ T_{\text{initial}} = 15.0 + 273.15 = 288.15 \, \text{K} \]
\[ T_{\text{melting}} = 1235 + 273.15 = 1508.15 \, \text{K} \]

Now, calculate ΔT:

\[ \Delta T = T_{\text{melting}} - T_{\text{initial}} = 1508.15 - 288.15 = 1220 \, \text{K} \]

**Hint:** Remember to convert Celsius to Kelvin by adding 273.15.

### Step 4: Calculate the energy required to heat the silver to its melting point (Q1)
Using the formula \( Q = mc\Delta T \):

\[ Q_1 = M \cdot c \cdot \Delta T \]
\[ Q_1 = 0.085 \, \text{kg} \cdot 236 \, \text{J/(kg·K)} \cdot 1220 \, \text{K} \]
\[ Q_1 = 0.085 \cdot 236 \cdot 1220 = 25,000.2 \, \text{J} \]

**Hint:** Make sure to multiply the mass, specific heat, and temperature change correctly.

### Step 5: Determine the latent heat of fusion for silver
The latent heat of fusion (L) for silver is approximately 10500 J/kg.

**Hint:** Latent heat is the amount of heat required to convert a unit mass of a solid into a liquid without a change in temperature.

### Step 6: Calculate the energy required to melt the silver (Q2)
Using the formula \( Q = mL \):

\[ Q_2 = M \cdot L \]
\[ Q_2 = 0.085 \, \text{kg} \cdot 10500 \, \text{J/kg} = 893.5 \, \text{J} \]

**Hint:** Ensure you use the correct units for latent heat.

### Step 7: Calculate the total energy required (Q_total)
The total energy required to completely melt the silver is the sum of Q1 and Q2:

\[ Q_{\text{total}} = Q_1 + Q_2 \]
\[ Q_{\text{total}} = 25000.2 \, \text{J} + 893.5 \, \text{J} = 25893.7 \, \text{J} \]

**Hint:** Add the two energy values together to find the total energy required.

### Final Answer
The minimum amount of energy required to completely melt 85.0 g of silver initially at 15.0 °C is approximately **25893.7 J**.