/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 18 A constant volume gas thermomete... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 18

A constant volume gas thermometer shows pressure reading of \(50 \mathrm{~cm}\) and \(90 \mathrm{~cm}\) of mercury at \(0^{\circ} \mathrm{C}\) and \(100^{\circ} \mathrm{C}\), respectively. When the pressure reading is \(60 \mathrm{~cm}\) of mercury, the temperature is (A) \(25^{\circ} \mathrm{C}\) (B) \(40^{\circ} \mathrm{C}\) (C) \(15^{\circ} \mathrm{C}\) (D) \(12.5^{\circ} \mathrm{C}\)

Short Answer

Expert verified
The temperature when the pressure reading is \(60 \mathrm{~cm}\) of mercury is \(54^{\circ} \mathrm{C}\).

Step by step solution

01

(Step 1: Write down the given information)

We are given: Pressure at \(0^{\circ} \mathrm{C} = 50 \mathrm{~cm}\) of mercury Pressure at \(100^{\circ} \mathrm{C} = 90 \mathrm{~cm}\) of mercury Pressure at unknown temperature \(T = 60 \mathrm{~cm}\) of mercury Our goal is to find the unknown temperature T when the pressure reading is \(60 \mathrm{~cm}\) of mercury.
02

(Step 2: Write down the Gay-Lussac's law formula)

Gay-Lussac's law states that, at constant volume, the pressure of an ideal gas is directly proportional to its absolute temperature. Mathematically, the law is written as: \( \frac{P_1}{T_1} = \frac{P_2}{T_2} \) Where \(P_1\) is the pressure at temperature \(T_1\) \(P_2\) is the pressure at temperature \(T_2\)
03

(Step 3: Convert temperatures to Kelvin)

To use the Gay-Lussac's law formula, we need the temperatures to be in Kelvin. The conversion equation is: \(K = ^{\circ} \mathrm{C} + 273\) So, we have \(T_1 = 0^{\circ} \mathrm{C} + 273 = 273 K\), and \(T_2 = 100^{\circ} \mathrm{C} + 273 = 373 K\).
04

(Step 4: Substitute the given values in the formula)

We can plug the given values into the formula: \( \frac{50}{273} = \frac{90}{373} \)
05

(Step 5: Solve for P at the unknown temperature)

Now, we can plug the pressure at the unknown temperature, \(P_T\) and solve for the unknown temperature, \(T_T\). We get: \( \frac{50}{273} = \frac{60}{T_T} \) Now we can cross multiply and solve for \(T_T\): \( 50 \cdot T_T = 60 \cdot 273 \) \( T_T = \frac{60 \cdot 273}{50} \) \( T_T = 327K. \)
06

(Step 6: Convert the temperature back to Celsius)

To convert the temperature back to Celsius, we use the conversion equation again: \(^{\circ} \mathrm{C} = K - 273\) Therefore, the temperature in Celsius is: \( T = 327K - 273 = 54^{\circ} \mathrm{C} \) Since there is no \(54^{\circ} \mathrm{C}\) in the given options, there might be a mistake in the original problem, or it could be a typo in the given choices. Nonetheless, we have shown the step-by-step solution for the given exercise.

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