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Chapter 18: Problem 42

In one of your rigorous workout sessions, you lost \(150 \mathrm{~g}\) of water through evaporation. Assume that the amount of work done by your body was \(1.80 \cdot 10^{5} \mathrm{~J}\) and that the heat required to evaporate the water came from your body. a) Find the loss in internal energy of your body, assuming the latent heat of vaporization is \(2.42 \cdot 10^{6} \mathrm{~J} / \mathrm{kg}\). b) Determine the minimum number of food calories that must be consumed to replace the internal energy lost (1 food calorie \(=4186\) J).

Short Answer

Expert verified
Answer: 44 food calories.

Step by step solution

01

Calculate the energy required to evaporate the given amount of water

To calculate the energy required to evaporate the given amount of water, we use the formula: Energy = Mass x Latent heat of vaporization where Mass = \(150\,\text{g}\) (converted to kg) and Latent heat of vaporization = \(2.42 \cdot 10^6\, \mathrm{J} / \mathrm{kg}\). Converting the mass of water to kg: \(150\,\text{g} \times \frac{1\,\text{kg}}{1000\, \text{g}} = 0.15\, \text{kg}\). Energy = \(0.15\,\text{kg} \times 2.42 \cdot 10^6 \,\mathrm{J} / \mathrm{kg}\) Energy = \(3.63 \cdot 10^5\, \text{J}\)
02

Determine the loss in internal energy

According to the conservation of energy, the work done by the body is equal to the energy required to evaporate the water plus the change in the internal energy of your body, Work done = Energy + Loss in internal energy Rearranging the equation, we get: Loss in internal energy = Work done - Energy Loss in internal energy = \(1.80 \cdot 10^5\, \text{J} - 3.63 \cdot 10^5\, \text{J}\) Loss in internal energy = \(-1.83 \cdot 10^5\, \text{J}\) The negative sign indicates that the internal energy of your body has decreased. b) Determine the minimum number of food calories
03

Convert the Joules to food calories

To convert the lost energy from Joules to food calories, we use the conversion factor given, 1 food calorie = 4186 J Lost energy in food calories = \(\frac{-1.83 \cdot 10^5\, \text{J}}{4186\, \mathrm{J} / \mathrm{food\, calorie}}\) Lost energy in food calories = \(-43.77\, \text{food calories}\)
04

Find the minimum number of food calories needed to replace the lost internal energy

Since energy cannot be negative, we must consume at least 43.77 food calories to replace the lost internal energy. Minimum number of food calories = 44 (rounded up)

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