# Mass of ice to warm a liquid - do I have it set up correctly?

• Linus Pauling
In summary, to reach a final temperature of 33.0C in an insulated beaker with 0.250 kg of liquid water at 76.1C, 0.0906 kg of ice at -12.5C must be added. This is calculated using the specific heat of liquid water (4190 J/kg*K), the specific heat of ice (2100 J/kg*K), and the heat of fusion for water (334 kJ/kg). The equation Q4 = Q1 + Q2 + Q3 is used, where Q1 represents the warming of the ice, Q2 represents the melting of the ice, Q3 represents bringing the liquids to the same temperature, and Q4 represents cooling the liquid to
Linus Pauling
1. An insulated beaker with negligible mass contains liquid water with a mass of 0.250 kg and a temperature of 76.1C.

How much ice at a temperature of -12.5C must be dropped into the water so that the final temperature of the system will be 33.0C? Take the specific heat of liquid water to be 4190 J/kg*K, the specific heat of ice to be 2100 J/kg*K, and the heat of fusion for water to be 334 kJ/kg.

2. Q = Mc deltaT
Q = +/- ML_f

3. warming the ice = Q1 = m_ice * c_ice * deltaT = m_ice * (2100) * (0 - 260.5K)

melting ice = Q2 = m_ice * L_f = m_ice * (334000 J/kg)

bring liquids to same temp. = Q3 = m_ice * c_water * deltaT = m_ice * 4190 * (306K - 0K)

cool water = Q4 = m_water * c_water * deltaT = 0.250 * 4190 * (306K - 349.1K)

Then, I did Q4 = Q1 + Q2 + Q3, and solved for m_ice:

-45147.25 = m_ice * (1282140 + 334000 - 547050)

m_ice = -0.0422, which I just took to be positive. I think I am at least on the right track, but have I set it up wrong since I get a negative number for mass?

Actually, I have to correct myself: I set the Q of cooling the liquid to final temperature equal to the other three (warming ice + melting ice + warming the new mass of liquid). Solving, I obtain the mass of the original ice to be 0.0906 kg.

Got it! Nevermind.

I would first like to commend you for setting up the problem correctly and using the appropriate equations and values for specific heat and heat of fusion. However, I do see a mistake in your calculations that is leading to the negative mass value.

Firstly, when calculating the heat required to warm the ice, you have used a temperature difference of 0-260.5K, which is incorrect. The correct temperature difference should be 33.0C - (-12.5C), which is 45.5K. This will change the value of Q1 to be -94500 J.

Secondly, when calculating the heat required to bring the liquids to the same temperature, you have used a temperature difference of 306K - 0K, which is also incorrect. The correct temperature difference should be 306K - 33.0C, which is 273K. This will change the value of Q3 to be 0.250 * 4190 * 273 = 283747.5 J.

Using these corrected values in your equation Q4 = Q1 + Q2 + Q3, you will get a positive value for m_ice, which is the correct result.

Therefore, the correct mass of ice that must be added to the system is approximately 0.063 kg. I hope this helps clarify your calculations. Keep up the good work!

## 1. How do I determine the mass of ice needed to warm a liquid?

The mass of ice needed to warm a liquid can be calculated using the specific heat capacity and enthalpy of fusion of water. The formula is: mass of ice = (enthalpy of fusion x mass of liquid) / specific heat capacity of ice. Make sure to use consistent units for all values.

## 2. Should I use the mass of the entire liquid or just the amount needed for the experiment?

You should only use the mass of the liquid needed for the experiment. This is because the mass of the entire liquid may include impurities or other substances that can affect the results.

## 3. Can I use any type of ice to warm a liquid?

Ideally, you should use pure water ice as it has a known specific heat capacity and enthalpy of fusion. Other types of ice, such as ice with impurities, may have different values and affect the accuracy of your results.

## 4. Is it necessary to account for the temperature of the ice when calculating the mass needed?

Yes, you should take into account the initial temperature of the ice when calculating the mass needed. This is because the ice will need to warm up to 0 degrees Celsius before it can start melting and warming the liquid.

## 5. Can I use this calculation for any type of liquid?

No, this calculation is specifically for water-based liquids. Other liquids may have different specific heat capacities and enthalpies of fusion, so the formula would need to be adjusted accordingly.

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