To determine the amount of heat required to raise the temperature of an 86.0 g sample of ethanol from 108.0 K to 300.8 K, let's follow these steps:
1. Given Data:
- Mass of ethanol, [tex]\( m = 86.0 \)[/tex] g
- Initial temperature, [tex]\( T_{initial} = 108.0 \)[/tex] K
- Final temperature, [tex]\( T_{final} = 300.8 \)[/tex] K
- Specific heat capacity, [tex]\( c = 242 \)[/tex] [tex]\( \text{J} / (\text{g} \cdot ^\circ \text{C}) \)[/tex]
2. Determine the temperature change:
[tex]\[
\Delta T = T_{final} - T_{initial}
\][/tex]
[tex]\[
\Delta T = 300.8 \, \text{K} - 108.0 \, \text{K} = 192.8 \, \text{K}
\][/tex]
(Note: Since the temperatures are given in Kelvin and we're considering only temperature differences, the units for [tex]\( \Delta T \)[/tex] in Kelvin are equivalent to degrees Celsius for the purpose of this calculation.)
3. Convert the specific heat capacity to kJ/g°C:
[tex]\[
c_{kJ} = 242 \, \text{J} / (\text{g} \cdot ^\circ \text{C}) \times 0.001 \, \text{kJ/J} = 0.242 \, \text{kJ} / (\text{g} \cdot ^\circ \text{C})
\][/tex]
4. Calculate the heat required using the formula [tex]\( Q = mc\Delta T \)[/tex]:
[tex]\[
Q = m \times c_{kJ} \times \Delta T
\][/tex]
[tex]\[
Q = 86.0 \, \text{g} \times 0.242 \, \text{kJ} / (\text{g} \cdot ^\circ \text{C}) \times 192.8 \, ^\circ \text{C}
\][/tex]
5. Perform the multiplication:
[tex]\[
Q = 86.0 \, \text{g} \times 0.242 \, \text{kJ} / (\text{g} \cdot ^\circ \text{C}) \times 192.8 \, ^\circ \text{C}
\][/tex]
[tex]\[
Q = 4012.5536 \, \text{kJ}
\][/tex]
Therefore, the amount of heat required is approximately [tex]\( 4012.5536 \)[/tex] kJ. Comparing this value with the options provided, none of them match exactly. Therefore, it seems there might be a misunderstanding regarding the results provided or possibly a typographical error in the options.
