Question 640: Thermodynamics & KTG

Question

A calorimeter contains 10 g of water at $20^{\circ} \mathrm{C}$. The temperature fall to $15^{\circ} \mathrm{C}$ in 10 min . When calorimeter contains 20 g of water at $20^{\circ}  \mathrm{C}$, it takes 15 min . for the temperature to become $15^{\circ} \mathrm{C}$. The water equivalent at the calorimeter is

Accepted Answer

Let the water equivalent of the calorimeter be $ x $ grams. This means the calorimeter acts like it contains $ x $ grams of water for heat calculations.
Formula:
The way the temperature of water falls is given by:

$ m\,c\,\frac{dT}{dt} = -k\left[\frac{T_1+T_2}{2} - T_o\right] $

where $ m $ is the total mass (water + calorimeter equivalent), $ c $ is specific heat, $ T_1 $ and $ T_2 $ are starting and ending temperatures, and $ T_o $ is the air temperature outside.
First case (10 g water):
$ (10+x)c \frac{(20-15)}{10} = -k\left[\frac{20+15}{2} - T_o\right] $
Second case (20 g water):
$ (20+x)c \frac{(20-15)}{15} = -k\left[\frac{20+15}{2} - T_o\right] $
The bracketed factor on the right is the same in both cases. The left side is different because of the different amounts of water and time.
Setting the left sides as equal ratios:
$ \frac{10+x}{10} = \frac{20+x}{15} $
Cross-multiplied, this gives:

$ 15(10+x) = 10(20+x) $
Expand:

$ 150 + 15x = 200 + 10x $
Move $ 10x $ to the left and $ 150 $ to the right:

$ 15x - 10x = 200 -150 $
$ 5x = 50 $
So, $ x = 10 $ grams.
The water equivalent of the calorimeter is 10 grams.

Answer

50 g

Answer

25 g

Answer

10 g

Answer

5 g

Thermodynamics & KTG

A calorimeter contains 10 g of water at 20^{\circ} \mathrm{C}. The temperature fall to 15^{\circ} \mathrm{C} in 10 min . When calorimeter contains 20 g of water at 20^{\circ} \mathrm{C}, it takes 15 min . for the temperature to become 15^{\circ} \mathrm{C}. The water equivalent at the calorimeter is

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