Heat and Internal Energy and Heat and Temperature Change: Specific Heat Capacity

In summary, to take a bath with the water temperature at 35.0 degrees C, you need to mix 49.0 degrees C water with 12.0 degrees C water to get the desired temperature.
  • #1
Axoren
17
0

Homework Statement


When you take a bath, how many kilograms of hot water (49°C) must you mix with cold water (12°C) so that the temperature of the bath is 36°C? The total mass of water (hot plus cold) is 191 kg. Ignore any heat flow between the water and its external surroundings.

Temperature of the Cold Water: 12 C
Temperature of the Hot Water: 49 C
Temperature of the Total Water: 36 C
Mass of the Cold Water: y, 191-x
Mass of the Hotwater: x, 191-y
Mass of the Total Water: 191 kg

Homework Equations


Q =mcΔT
Where Q is Joules of Energy, m is Mass of the substance (kg), c is the specific heat coefficient, ΔT is change in Temperature (in Celsius)

The Attempt at a Solution


Total Q = 191 kg * 4186 J/kgC * 36 C
Total Q = 28782936 J
Hot Q = X kg * 4186 J/kgC * 13 C
Cold Q = (191 - X) kg * 4186 J/kgC * 24 C
28782936 J = Hot Q + Cold Q
28782936 J = (X kg * 4186 J/kgC * 13 C) + ((191 - X) kg * 4186 J/kgC * 24 C)
28782936 J = (X kg * 54418 J/kg) + ((191 - X) kg * 100464 J/kg)
28782936 J = 54418X J + 19188624 J - 100464X J
28782936 J = 19188624 J - 46046X J
9594312 J = 46046X J
208.36 kg = X (Answer only needs to be to 2 decimal places)

^This does not make sense, because the mass of the hot water is larger than the mass of the total water. Assistance?
 
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  • #2
Axoren said:

The Attempt at a Solution


Total Q = 191 kg * 4186 J/kgC * 36 C
Total Q = 28782936 J
Hot Q = X kg * 4186 J/kgC * 13 C
Cold Q = (191 - X) kg * 4186 J/kgC * 24 C
28782936 J = Hot Q + Cold Q
28782936 J = (X kg * 4186 J/kgC * 13 C) + ((191 - X) kg * 4186 J/kgC * 24 C)
28782936 J = (X kg * 54418 J/kg) + ((191 - X) kg * 100464 J/kg)
28782936 J = 54418X J + 19188624 J - 100464X J
28782936 J = 19188624 J - 46046X J
9594312 J = 46046X J
208.36 kg = X (Answer only needs to be to 2 decimal places)

^This does not make sense, because the mass of the hot water is larger than the mass of the total water. Assistance?
Total Q = 0 .

So the equation should be [itex]Q_h + Q_c=0[/itex]

AM
 
  • #3
Thank you very much! I don't know how I missed that.
 
  • #4
Hey I really need this answered I've been working on it for 3 hours help.
Question: You want to take a bath with the water temperature at 35.0 degrees C. The water temperature is 38 degrees C from the hot water tap and 11 degrees C from the cold water tap. You fill the tub with a total of 187 kg of water. How many kilograms of water from the hot water tap do you use?
Please help.
 
  • #5
Just kidding I'm stupid. i figured it out... sorry for wasting your time.
 

FAQ: Heat and Internal Energy and Heat and Temperature Change: Specific Heat Capacity

1. What is the difference between heat and internal energy?

Heat and internal energy are related concepts, but they are not the same. Heat refers to the transfer of energy from a hotter object to a colder object. On the other hand, internal energy is the total energy of all the particles in a substance, including kinetic and potential energy. Heat can cause a change in internal energy, but they are not interchangeable terms.

2. How is specific heat capacity related to heat and temperature change?

Specific heat capacity is a measure of how much heat is needed to raise the temperature of a substance by a certain amount. It is directly related to heat and temperature change through the equation Q = mcΔT, where Q is the heat transferred, m is the mass of the substance, c is the specific heat capacity, and ΔT is the change in temperature. In other words, the specific heat capacity determines the amount of heat needed to cause a certain temperature change in a substance.

3. What factors affect the specific heat capacity of a substance?

The specific heat capacity of a substance is affected by its chemical composition and physical properties such as density, state (solid, liquid, gas), and temperature. Generally, substances with higher molecular complexity or stronger intermolecular forces have higher specific heat capacities. Additionally, the specific heat capacity of a substance may vary with temperature, especially for gases.

4. How is specific heat capacity measured experimentally?

To measure the specific heat capacity of a substance, an experimental setup called a calorimeter is used. The substance is placed in the calorimeter, and its initial and final temperatures are recorded after a known amount of heat is added or removed. Using the equation Q = mcΔT, the specific heat capacity can be calculated by rearranging the equation to solve for c.

5. What are some real-life applications of specific heat capacity?

Specific heat capacity has many practical applications, including cooking, heating and cooling systems, and thermal insulation. For example, the specific heat capacity of water is high, making it an ideal substance to use for cooling purposes. In heating and cooling systems, the specific heat capacity of materials is important in determining how much energy is needed to maintain a desired temperature. In thermal insulation, materials with low specific heat capacities are used to prevent heat transfer, such as in the insulation of buildings.

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