You decide to use your body as a Carnot heat engine. The operating gas is in a tube with one end in your mouth (where the temperature is 37.0 C) and the other end at the surface of your skin, at 30.0 C.
What is the maximum efficiency of such a heat engine?
e=2.26%
Suppose you want to use this human engine to lift a 2.05 kg box from the floor to a tabletop 1.20 m above the floor. How much must you increase the gravitational potential energy?
W=24.1 J
Suppose you want to use this human engine to lift a 2.05 kg box from the floor to a tabletop 1.20 m above the floor. How much heat input is needed to accomplish this?
Q=1070 J
How many 350 calorie (those are food calories, remember) candy bars must you eat to lift the box in this way? Recall that 80.0 % of the food energy goes into heat.
??? candy bars
*I know that it will be a very small number, but I'm not quite sure how to calculate it!
Carnot efficiency = 1 - 303/310 = 2.26% OK so far
W = 2.05 kg*9.8 m/s^2*1.2 m = 24.1 J Right again.
0.0226Q = 24.1 J
Q = 1066 J close enough
1066 J is 255 calories but that is 0.255 Kcal, or Food Calories. Each candy bar is 350,000 (gram)-calories. Eating one bar produces 0.8*350,000 cal of heat or 280,000 cal.
You only need to consume 255/280,000 = 0.0009 of the candy bar.
This is a self-contradicting problem because in the beginning it says to treat the body as a 2.26% efficient engine but in the end it says to assume 20% efficiency (80% going to heat) when calculating the candy bar requirement. The body is able to extract more than Carnot efficiency from food because it is NOT a heat engine. There are other less entropy-producing ways to convert chemical energy to work - electrochemical processes. That is how fuel cells work
Well, if you're using your body as a heat engine, you might want to consider eating some clown-sized candy bars to keep your energy up! Let's see how many you'll need.
First, let's convert the heat input needed to lift the box, which is 1070 J, into food calories. Since 80% of the food energy goes into heat, we can divide the heat input by 0.8 to get the food energy needed.
1070 J / 0.8 = 1337.5 calories
Now, since each candy bar has 350 calories, we can divide the food energy needed by the calories in one candy bar to find out how many you'll need.
1337.5 calories / 350 calories per candy bar = 3.822 candy bars
So, you would need approximately 3.822 candy bars to lift the box using your human engine. But of course, it's important to remember to have a balanced diet and not rely solely on candy bars for your energy needs!
To calculate the number of candy bars needed, we need to find the energy input required to lift the box and determine how many calories are equivalent to that energy input.
First, let's find the energy input required to lift the box using the formula:
W = mgh
where W is the work done (energy input), m is the mass of the box, g is the acceleration due to gravity (9.8 m/s^2), and h is the height the box is lifted.
W = (2.05 kg) * (9.8 m/s^2) * (1.20 m)
W = 23.8 J
Next, let's calculate the energy equivalent in calories. Since 80% of the food energy goes into heat, we need to divide the energy input by 80%:
Q = W / 0.8
Q = 23.8 J / 0.8
Q = 29.8 J
To convert the energy from joules to calories, we'll use the conversion factor: 1 calorie = 4.184 joules.
Q_in_calories = Q / 4.184
Q_in_calories = 29.8 J / 4.184
Q_in_calories = 7.12 calories
Finally, let's find the number of candy bars needed, assuming each candy bar contains 350 calories:
number_of_candy_bars = Q_in_calories / 350
number_of_candy_bars = 7.12 calories / 350
number_of_candy_bars ≈ 0.02 candy bars (rounded to two decimal places)
Therefore, you would need to eat approximately 0.02 candy bars (or 1/50th of a candy bar) to lift the box using this human engine.
To calculate the number of candy bars you would need to eat, we first need to determine the total energy required to lift the box, and then calculate how many candy bars it would take to generate that amount of energy.
The work done in lifting the box can be calculated using the equation:
W = m * g * h
where:
W is the work done (in joules),
m is the mass of the object (in kilograms),
g is the acceleration due to gravity (approximately 9.8 m/s^2),
h is the height the object is lifted (in meters).
In this case, the mass of the box (m) is given as 2.05 kg and the height (h) as 1.20 m. Plugging these values into the equation, we get:
W = 2.05 kg * 9.8 m/s^2 * 1.20 m
Calculating this gives us:
W ≈ 24.1 J
So the total energy required to lift the box is approximately 24.1 joules.
Now, we need to consider the efficiency of the human engine. The Carnot efficiency is given by the equation:
e = 1 - (Tc / Th)
where:
e is the efficiency,
Tc is the temperature at which heat is rejected (our skin temperature of 30.0°C),
Th is the temperature at which heat is absorbed (our body temperature of 37.0°C).
Plugging in the values, we get:
e = 1 - (30.0°C + 273.15) / (37.0°C + 273.15)
Calculating this gives us:
e ≈ 0.0226 or 2.26%
This means that only 2.26% of the heat input would be converted into useful work, with the remaining 97.74% being rejected as waste heat.
Next, we need to convert the total energy required (24.1 joules) into the equivalent food energy. We know that 80% of the food energy goes into heat, so we need to account for this factor.
Since we are looking for the number of candy bars needed, we can use the energy content of a single candy bar as a reference. The energy content is given as 350 calories per candy bar, where 1 calorie is equivalent to 4.184 joules.
To calculate the number of candy bars needed, we can use the equation:
Number of candy bars = (Total energy required / Food energy per candy bar) / Food energy conversion factor
Plugging in the values, we get:
Number of candy bars = (24.1 J / (350 cal * 4.184 J/cal)) / 0.8
Calculating this gives us:
Number of candy bars ≈ 0.017
Therefore, you would need to eat approximately 0.017 or less than one candy bar to generate enough energy to lift the box using this human engine.
Note: Keep in mind that this calculation assumes 100% efficiency in converting food energy to heat, which may not be the case in reality due to various factors like digestion and metabolism.