A researcher studying the nutritional value of a new candy places a 4.70-gram sample of the candy inside a bomb calorimeter and combusts it in excess oxygen. The observed temperature increase is 2.98 °C. If the heat capacity of the calorimeter is 39.80 kJ·K–1, how many nutritional Calories are there per gram of the candy?
To calculate the number of nutritional Calories (kcal) per gram of the candy, we need to determine the heat energy released during the combustion.
1. Convert the observed temperature increase from Celsius to Kelvin:
ΔT = 2.98 °C = 2.98 K
2. Use the heat capacity of the calorimeter to find the amount of heat energy released:
q = CΔT
= (39.80 kJ·K–1)(2.98 K)
3. Convert the heat energy from kilojoules to joules:
q = (39.80 kJ·K–1)(2.98 K) × 1000 J/kJ
= (119 kJ)
4. Calculate the quantity of heat energy released per gram of the candy:
q = 119 kJ
m = 4.70 g
Heat Energy per gram = q/m
= (119 kJ)/(4.70 g)
5. Convert the heat energy from joules to nutritional Calories:
1 kcal = 4.184 kJ
Nutritional Calories per gram of the candy = (119 kJ)/(4.70 g) × (1 kcal/4.184 kJ)
To calculate the nutritional Calories per gram of the candy, we need to find the amount of heat energy released during combustion.
The heat energy released during combustion can be calculated using the formula:
q = C * ΔT
where:
q is the heat energy (in joules)
C is the heat capacity of the calorimeter (in joules per degree Celsius or Kelvin)
ΔT is the temperature increase (in degrees Celsius or Kelvin)
First, we need to convert the temperature increase from Celsius to Kelvin:
ΔT = 2.98 °C + 273.15 = 276.13 K
Now we can calculate the heat energy released:
q = (39.80 kJ·K^(-1)) * (276.13 K)
q = 10,992.774 kJ
Next, we need to convert the heat energy from kilojoules (kJ) to calories (cal). One calorie is equal to 4.184 joules:
10,992.774 kJ * (4.184 kJ/cal) = 45,997.423 cal
Finally, we need to find the nutritional Calories per gram of the candy. A nutritional Calorie is defined as a kilocalorie (kcal) or 1,000 calories:
45,997.423 cal / 4.70 g = 9,785.186 kcal/g
Therefore, there are approximately 9,785.186 nutritional Calories per gram of the candy.