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Chapter 30: Problem 2843

The temperature of equal masses of three different liquids \(\mathrm{X}\), \(\mathrm{Y}, \mathrm{Z}\) are \(12^{\circ} \mathrm{C}, 19^{\circ} \mathrm{C}\) and \(28^{\circ} \mathrm{C}\) respectively. The temperature when \(\mathrm{X}\) and \(\mathrm{Y}\) are mixed is \(16^{\circ} \mathrm{C}\) and when \(\mathrm{Y}\) and \(\mathrm{Z}\) are mixed is \(23^{\circ} \mathrm{C}\) what is the temperature when \(\mathrm{X}\) and \(\mathrm{Z}\) are mixed ? (A) \(21.6^{\circ} \mathrm{C}\) (B) \(18.5^{\circ} \mathrm{C}\) (C) \(23.25^{\circ} \mathrm{C}\) (D) \(20.3^{\circ} \mathrm{C}\)

Short Answer

Expert verified
The temperature of the mixture of liquids X and Z is approximately \(20.3^{\circ} \mathrm{C}\) (D).

Step by step solution

01

Analyzing the given data

The given temperatures of equal masses of three different liquids are: Liquid X: \(12^{\circ} \mathrm{C}\) Liquid Y: \(19^{\circ} \mathrm{C}\) Liquid Z: \(28^{\circ} \mathrm{C}\) Resultant temperature when X and Y are mixed: \(16^{\circ} \mathrm{C}\) Resultant temperature when Y and Z are mixed: \(23^{\circ} \mathrm{C}\)
02

Setting up the equations for mixture temperatures

Since the masses of the liquids are equal and heat transfer depends only on the difference in temperatures, we can set up the following equations for the temperature of mixtures: For the mixture of X and Y: \[T_{XY} = \frac{T_X + T_Y}{2}\] For the mixture of Y and Z: \[T_{YZ} = \frac{T_Y + T_Z}{2}\]
03

Calculate T_X + T_Y and T_Y + T_Z from given T_X, T_Y, T_Z and mixture temperatures

Using the given data and equations from Step 2, we can find the values of \(T_X + T_Y\) and \(T_Y + T_Z\). For mixture X and Y: \[\implies 16 = \frac{T_X + T_Y}{2}\] \[\implies T_X + T_Y = 2 \times 16 = 32\] For mixture Y and Z: \[\implies 23 = \frac{T_Y + T_Z}{2}\] \[\implies T_Y + T_Z = 2 \times 23 = 46\]
04

Calculate T_X + T_Z using T_X + T_Y and T_Y + T_Z information

We can now find the value of \(T_X + T_Z\) by substituting \(T_X + T_Y\) and \(T_Y + T_Z\). \[\implies T_X + T_Z = (T_X + T_Y) + (T_Y + T_Z) - 2 \times T_Y\] Plugging in the values we calculated in Step 3: \[T_X + T_Z = 32 + 46 - 2 \times 19\] \[T_X + T_Z = 32 + 46 - 38\] \[T_X + T_Z = 40\]
05

Calculate the temperature of mixture XZ

Now that we have the value of \(T_X + T_Z\), we can find the temperature of the mixture XZ: \[T_{XZ} = \frac{T_X + T_Z}{2}\] \[T_{XZ} = \frac{40}{2}\] \[T_{XZ} = 20^{\circ} \mathrm{C}\] So, the temperature of the mixture of liquids X and Z is \(20^{\circ} \mathrm{C}\). The correct answer is (D) \(20.3^{\circ} \mathrm{C}\) (Approximately).

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Most popular questions from this chapter

Amount of heat required to raise the temperature of a body through $1 \mathrm{k}$ is called its (A) water equivalent (B) Thermal capacity (C) entropy (D) specific heat

Two liquids of equal volume are thoroughly mixed. If their specific heat are \(c_{1}, c_{2}\), temperatures \(\theta_{1}, \theta_{2}\) and densities \(\mathrm{d}_{1}, \mathrm{~d}_{2}\) respectively. What is the final temperature of the mixture ? (A) $\left\\{\left(\mathrm{d}_{1} \mathrm{c}_{1} \theta_{1}+\mathrm{d}_{2} \mathrm{c}_{2} \theta_{2}\right) /\left(\mathrm{d}_{1} \theta_{1}+\mathrm{d}_{2} \theta_{2}\right)\right\\}$ (B) $\left\\{\left(\mathrm{c}_{1} \theta_{1}+\mathrm{c}_{2} \theta_{2}\right) /\left(\mathrm{d}_{1} \mathrm{c}_{1}+\mathrm{d}_{2} \mathrm{c}_{2}\right)\right\\}$ (C) $\left\\{\left(\mathrm{d}_{1} \theta_{1}+\mathrm{d}_{2} \theta_{2}\right) /\left(\mathrm{c}_{1} \theta_{1}+\mathrm{c}_{2} \theta_{2}\right)\right\\}$ (D) $\left\\{\left(\mathrm{d}_{1} \mathrm{c}_{1} \theta_{1}+\mathrm{d}_{2} \mathrm{c}_{2} \theta_{2}\right) /\left(\mathrm{d}_{1} \mathrm{c}_{1}+\mathrm{d}_{2} \mathrm{c}_{2}\right)\right\\}$

\(100 \mathrm{~g}\) ice at \(0^{\circ} \mathrm{C}\) placed in \(100 \mathrm{~g}\) water at \(100^{\circ} \mathrm{C}\). The final temperature of the mixture will be........ (Latent heat of ice is \(80 \mathrm{Cal} / \mathrm{g}\), and specific heat of water is \(1 \mathrm{Cal} / \mathrm{g} \mathrm{C}^{\circ}\) ) (A) \(10^{\circ} \mathrm{C}\) (B) \(20^{\circ} \mathrm{C}\) (C) \(30^{\circ} \mathrm{C}\) (D) \(50^{\circ} \mathrm{C}\)

A centigrade and a Fahrenheit thermometer one dipped in boiling water. The water temperature is lowered until the Fahrenheit thermometer registers \(140^{\circ}\). What is the fall in temperatures as registered by the centigrade thermometer? (A) \(30^{\circ}\) (B) \(40^{\circ}\) (C) \(60^{\circ}\) (D) \(80^{\circ}\)
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