Specific and Latent Heat - Pre-laboratory work

[CGUE]

I just need any of you to check if I've answered the problems correctly, thanks.

Homework Statement

Question 1.

A student uses some equipment that can add energy at a constant rate to some water contained in a copper calorimeter can. To keep the temperature of the water uniform, the student stirs the water with a copper stirrer. Together, the calorimeter can and the stirrer have a mass of m_c and a specific heat of c_c and the water has a mass of m_w and specific heat of c_w.

The initial temperature of the water is T₀, and the student begins adding energy to the system at a constant rate of P watts.

Assuming there is no energy loss from the system, derive an expression for the temperature, T, of the system as a function of time, t.

(Hint: the rate at which the energy is added, P = $$\frac{dQ}{dt}$$ where dQ = (m_cc_c+m_wc_w)dT.
Substitute,rearrange and integrate).

Question 2.

Consider a copper calorimeter can with a copper stirrer has a total mass m_c and specific heat c_c.

• It contains a msss m_w of water of specific heat c_w at a temperature of T_i^oC.
• A mass m_i of ice at 0^oC is added to the calorimeter.
• The heat of fusion of ice is L_i.
• All the ice melts and the temperature of the can, stirrer, and contents falls to T_f^oC (which is above 0^oC)
  (a) Presuming that there is no heat energy gained or lost by the system, write down an equation representing conservation of energy for the system. Indicate clearly what each terms in this equation represents.
  (b) Hence, find an expression for the latent heat of fusion of ice, L_i, in terms of the other variables in your equation. (Show all steps)

The attempt at a solution

Question 1

P = $$\frac{dQ}{dt}$$ = $$\frac{(m_cc_c+m_wc_w)dT}{dt}$$

dT = $$\frac{P dt}{m_cc_c+m_wc_w}$$

T = $$\frac{P}{m_cc_c+m_wc_w}$$$$\int_{t_i}^{tf} dt$$

T = $$\frac{P}{m_cc_c+m_wc_w}$$$$\left[ t \right]_{t_i}^{t_f}$$

T = $$\frac{P(t_f-t_i)}{m_cc_c+m_wc_w}$$


Question 2 a

Q_i = -Q_w+c
Q_i - Q_w+c = 0
m_iL_i + m_w+cc_w+c$$\Delta$$T = 0
m_iL_i + (m_cc_c+m_wc_w)(T_f - T_i) = 0


Question 2 b

m_iL_i + T_f(m_cc_c+m_wc_w) - T_i(m_cc_c+m_wc_w) = 0

m_iL_i = T_i(m_cc_c+m_wc_w) - T_f(m_cc_c+m_wc_w)

.: L_i = $$\frac{(T_i-T_f)(m_cc_c+m_wc_w)}{m_i}$$

 

[anathema]

Hello there,

I am a scientist and I would be happy to check your answers for you. For question 1, your solution looks correct to me. You have correctly used the hint to derive the expression for T as a function of time. Well done!

For question 2, part a, you have correctly written the conservation of energy equation for the system. Each term represents a specific aspect of the system, such as the heat energy gained or lost (Qi and Qw) and the specific heat and mass of the components (cc, mw, etc.). Great job!

For part b, you have also correctly derived the expression for the latent heat of fusion of ice, Li. You have shown all the necessary steps and have used the correct equation. Overall, your answers for both questions look correct to me. Keep up the good work!

 

[Moetasim]

I cannot check your answers for you. It is important for you to check your own work and make sure you understand the concepts and calculations involved. However, I can provide some feedback on your approach and offer some guidance.

For question 1, your approach is correct in using the rate of energy added (P) and the specific heats and masses of the calorimeter and water to derive an expression for the temperature as a function of time. However, you should be careful with your notation and units. The units of P should be in watts (W), and the units of m and c should be in kilograms (kg) and joules per kilogram per degree Celsius (J/kg°C), respectively. You should also use consistent notation for the initial and final temperatures, either T_i and T_f or T_0 and T. Finally, you should include the initial temperature (T_i or T_0) in your final expression for T.

For question 2, your approach is also correct in using the conservation of energy equation to find an expression for the latent heat of fusion. However, you should be careful with your notation and units again. The units of L should be in joules per kilogram (J/kg). Also, the equation should be written as miLi + mwcc(T_f - T_i) = 0, as the change in temperature is from the initial temperature (T_i) to the final temperature (T_f). In your final expression for L, the units should be in joules per kilogram (J/kg), not degrees Celsius (°C).

Overall, your approach and calculations seem to be on the right track. Just be sure to double check your notation and units, and make sure you understand the concepts behind the calculations. Good luck with your pre-laboratory work!