Q35P An insulated Thermos contains 13... [FREE SOLUTION]

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Chapter 18: Q35P (page 542)

An insulated Thermos contains $130 {cm}^{3}$ of hot coffee at $80 .0 掳颁$ .You put in a $12 .0 g$ ice cube at its melting point to cool the coffee. By how many degrees has your coffee cooled once the ice has melted and equilibrium is reached? Treat the coffee as though it were pure water and neglect energy exchanges with the environment.

Short Answer

Expert verified

The temperature when the ice has melted down and equilibrium is reached is $13.5 掳 C$

Step by step solution

Step 1: Identification of given data

a) Mass of coffee, $m_{w} = 130 鈥塯(1 鈥塯 = 1 {鈥塩尘}^{3})$

b) Mass of ice, $m_{I} = 12 鈥塯$

c) Initial temperature, $T_{i} = 80.0 掳 C$

d) Temperature of ice melting is $0.0 掳 C$

e) Specific heat of ice water, $c_{w} = 4190 鈥塉/办驳掳 C$

Significance of specific heat

The specific heat capacity is the amount heat required to raise the temperature of unit mass of substance by one degree.

In this process of heat absorption and release, the given mass of the ice is melted down to reach the state of equilibrium at a particular temperature, Using both the given formulas of heat absorption and heat released, we get the required temperature by the body to meet the expectations. Again, by the temperature difference of the found answer, we will get the required value.

Formulae:

The heat energy required by a body, $Q = m c 螖 T$ 鈥�(颈)

Here, $m$ = mass

$c$ = specific heat capacity

$螖 T$ = change in temperature

$Q$ = required heat energy

The heat energy required by a body, $Q = N c_{m} 螖 T$ 鈥�(颈颈)

Where, $N = m / M$ = number of moles

$c_{m}$ = molar specific heat capacity

$螖 T$ = change in temperature

Determining the temperature at which equilibrium is reached

Using equations (i) and (ii), we get the heat absorbed by the ice in the whole process of fusion as:

$Q_{1} = m_{I} c_{w} (T_{f} 鈭� 0 掳 C) + L_{f} m_{I}$ 鈥�(颈颈颈)

Here, $T_{f}$ is the latent heat of fusion.

Similarly, the heat transferred by the coffee is given using equation (i) as:

$Q_{2} = m_{w} c_{w} (T_{i} 鈭� T_{f})$ 鈥�(颈惫)

To reach the condition of equilibrium, the value of equation (iii) should be equal to equation (iv), so

$Q_{1} = Q_{2}$

To find the final temperature, the above setting can be written as:

$\begin{matrix} T_{f} = \frac{m_{w} c_{w} T_{I} 鈭� L_{f} m_{I}}{(m_{I} + m_{w}) c_{w}} \\ = [\frac{130 鈥塯脳 4190 鈥塉/办驳掳 C 脳 80 鈥� 掳 C 鈭� 333 脳 10^{3} J/g 脳 12鈥塯}{(12 鈥塯 + 130 鈥塯) (4190 鈥塉/办驳掳 C)}] 掳 C \\ = 66.5 掳 C \end{matrix}$