The amount of heat needed to melt 39.7 g of solid ethanol and bring it to a temperature of -24.2 °C is 9.60972 kJ.
To calculate the amount of heat needed to melt 39.7 g of solid ethanol and bring it to a temperature of -24.2 °C, we need to consider two steps: the heat required to melt the solid and the heat required to cool the liquid to -24.2 °C.
First, calculate the heat required to melt the solid ethanol using the heat of fusion. The heat of fusion for ethanol is 38.6 kJ/mol. Calculate the moles of ethanol using its molar mass (46.07 g/mol) and then multiply by the heat of fusion:
(39.7 g ethanol) / (46.07 g/mol) = 0.861 mol ethanol
Heat required to melt the solid = (0.861 mol ethanol) * (38.6 kJ/mol) = 33.2546 kJ
Next, calculate the heat required to cool the liquid ethanol to -24.2 °C using the specific heat capacity. The specific heat capacity of ethanol is 2.44 J/g·°C. Calculate the change in temperature:
Change in temperature = (-24.2 °C) - (0 °C) = -24.2 °C
Heat required to cool the liquid = (39.7 g ethanol) * (2.44 J/g·°C) * (-24.2 °C) / 1000 = -23.64488 kJ
Add the heats from both steps together to get the total heat required:
Total heat required = 33.2546 kJ + (-23.64488 kJ) = 9.60972 kJ