To determine the total amount of heat required to melt a [tex]$0.0500 \, \text{kg}$[/tex] (50 grams) ice cube starting at [tex]$-18.5^{\circ} \text{C}$[/tex], we need to perform the following steps:
1. Calculate the amount of heat required to raise the temperature of the ice from [tex]$-18.5^{\circ} \text{C}$[/tex] to [tex]$0^{\circ} \text{C}$[/tex].
2. Calculate the amount of heat required to melt the ice at [tex]$0^{\circ} \text{C}$[/tex] into water.
3. Sum the heats calculated in the previous steps to find the total heat required.
### Step 1: Heating the Ice to [tex]$0^{\circ} \text{C}$[/tex]
The formula to calculate the heat required to change the temperature of a substance is:
[tex]\[ q = m \cdot c \cdot \Delta T \][/tex]
Where:
- [tex]\( q \)[/tex] is the heat energy (in joules, [tex]\( J \)[/tex]).
- [tex]\( m \)[/tex] is the mass of the substance (in kilograms, [tex]\( kg \)[/tex]).
- [tex]\( c \)[/tex] is the specific heat capacity (in joules per kilogram per degree Celsius, [tex]\( J/(kg \cdot ^{\circ}C) \)[/tex]).
- [tex]\( \Delta T \)[/tex] is the change in temperature (final temperature - initial temperature).
For ice, the specific heat capacity [tex]\( c_{\text{ice}} = 2090 \, J/(kg \cdot ^{\circ}C) \)[/tex].
So, we need to find the heat required to raise the temperature of [tex]$0.0500 \, \text{kg}$[/tex] of ice from [tex]$-18.5^{\circ} \text{C}$[/tex] to [tex]$0^{\circ} \text{C}$[/tex]:
[tex]\[ \Delta T = 0^{\circ}C - (-18.5^{\circ}C) = 18.5^{\circ}C \][/tex]
[tex]\[ q_{\text{heat ice}} = 0.0500 \, \text{kg} \cdot 2090 \, J/(kg \cdot ^{\circ}C) \cdot 18.5^{\circ}C \][/tex]
[tex]\[ q_{\text{heat ice}} = 1933.25 \, J \][/tex]
### Step 2: Melting the Ice at [tex]$0^{\circ} \text{C}$[/tex]
The heat required to melt ice is determined by the latent heat of fusion [tex]\( L_{\text{fusion}} \)[/tex], which is the amount of heat required to change [tex]$1 \, \text{kg}$[/tex] of a solid into a liquid without a temperature change.
For water, [tex]\( L_{\text{fusion}} = 3.33 \times 10^5 \, J/kg \)[/tex].
The heat required to melt [tex]$0.0500 \, \text{kg}$[/tex] of ice is:
[tex]\[ q_{\text{melt ice}} = m \cdot L_{\text{fusion}} \][/tex]
[tex]\[ q_{\text{melt ice}} = 0.0500 \, \text{kg} \cdot 3.33 \times 10^5 \, J/kg \][/tex]
[tex]\[ q_{\text{melt ice}} = 16650.0 \, J \][/tex]
### Step 3: Total Heat Required
Finally, sum up the heat required to warm the ice to [tex]$0^{\circ} \text{C}$[/tex] and the heat required to melt it:
[tex]\[ q_{\text{total}} = q_{\text{heat ice}} + q_{\text{melt ice}} \][/tex]
[tex]\[ q_{\text{total}} = 1933.25 \, J + 16650.0 \, J \][/tex]
[tex]\[ q_{\text{total}} = 18583.25 \, J \][/tex]
Thus, the total amount of heat required to melt the [tex]$0.0500 \, \text{kg}$[/tex] ice cube completely, starting from [tex]$-18.5^{\circ} \text{C}$[/tex], is 18583.25 Joules.
