Chapter 1, Problem 67E

Use the figure below to answer the following questions.

1. a. Derive the relationship between °C and °X.
2. b. If the temperature outside is 22.0°C, what is the temperature in units of °X?
3. c. Convert 58.0°X to units of °C, K, and °F.

Interpretation Introduction

Interpretation:

The relationship between $\text{°C}$ and $\text{°X}$ should be derived and the given $\text{°C}$ should be converted in to $\text{°X}$ and the given $\text{°X}$ should be converted in to degree Celsius, Kelvin and Fahrenheit.

Concept introduction:

Conversion of temperature units in between degree Celsius, Kelvin and Fahrenheit are given by

$\begin{array}{l}{\text{Conversion of }}^{\text{o}}\text{F}\text{into}{}^{\text{o}}\text{C:}\\ {}^{\text{o}}\text{C}\text{=}\left({}^{\text{o}}\text{F}\text{-}{\text{32}}^{\text{o}}\text{F}\right)\text{×}\frac{{\text{5}}^{\text{o}}\text{C}}{{\text{9}}^{\text{o}}\text{F}}\\ {\text{Conversion of }}^{\text{o}}\text{C}\text{into}{}^{\text{o}}\text{F:}\\ {}^{\text{o}}\text{F}\text{=}\left({}^{\text{o}}\text{C}\text{×}\frac{{\text{9}}^{\text{o}}\text{F}}{{\text{5}}^{\text{o}}\text{C}}\right)\text{+}{\text{32}}^{\text{o}}\text{F}\\ {\text{Conversion of }}^{\text{o}}\text{C}\text{into}\text{K:}\\ \text{K = °C + 273}\text{.15}\end{array}$

Answer to Problem 67E

The relationship between $\text{°C}$ and $\text{°X}$ temperatures is,

$\text{°X=}\left(\frac{\text{5}}{\text{14}}\text{×°C}\right)\text{+}\frac{\text{50}}{\text{14}}$

Explanation of Solution

To derive the relationship between $\text{°C}$and $\text{°X}$ temperatures

Given,

Fig.1

In the given Fig.1,

In the given Fig.1, in thermometer (1) temperature difference is $\text{-10}\text{to}\text{130}\text{=}\text{140°C}$

In the given Fig.1, in thermometer (2) temperature difference is $\text{0}\text{to}\text{50}\text{=}\text{50}\text{°X}$

Hence equations for the inter conversion of $\text{°X}$ and $\text{°C}$ are,

Temperature $\text{°C}$ in to $\text{°X}$ is,

$\begin{array}{l}\text{=}\frac{\text{50}}{\text{140}}\text{=}\frac{\text{5}}{\text{4}}\\ \text{°X=}\frac{\text{5}}{\text{14}}\text{(°C)+}\frac{\text{50}}{\text{14}}\end{array}$

Temperature $\text{°X}$ in to $\text{°C}$ is,

$\begin{array}{l}\text{°C}\text{=}\frac{\text{14}}{\text{5}}\text{°X-}\frac{\text{50}}{\text{14}}\\ \text{°C=}\frac{\text{14}}{\text{5}}\text{°X-10}\end{array}$

Interpretation Introduction

Interpretation:

The relationship between $\text{°C}$ and $\text{°X}$ should be derived and the given $\text{°C}$ should be converted in to $\text{°X}$ and the given $\text{°X}$ should be converted in to degree Celsius, Kelvin and Fahrenheit.

Concept introduction:

Conversion of temperature units in between degree Celsius, Kelvin and Fahrenheit are given by

$\begin{array}{l}{\text{Conversion of }}^{\text{o}}\text{F}\text{into}{}^{\text{o}}\text{C:}\\ {}^{\text{o}}\text{C}\text{=}\left({}^{\text{o}}\text{F}\text{-}{\text{32}}^{\text{o}}\text{F}\right)\text{×}\frac{{\text{5}}^{\text{o}}\text{C}}{{\text{9}}^{\text{o}}\text{F}}\\ {\text{Conversion of }}^{\text{o}}\text{C}\text{into}{}^{\text{o}}\text{F:}\\ {}^{\text{o}}\text{F}\text{=}\left({}^{\text{o}}\text{C}\text{×}\frac{{\text{9}}^{\text{o}}\text{F}}{{\text{5}}^{\text{o}}\text{C}}\right)\text{+}{\text{32}}^{\text{o}}\text{F}\\ {\text{Conversion of }}^{\text{o}}\text{C}\text{into}\text{K:}\\ \text{K = °C + 273}\text{.15}\end{array}$

Answer to Problem 67E

The temperature $\text{22°C}$ is $\text{4}\text{.2 °X}$

Explanation of Solution

To convert the given temperature $\text{°X}$in to $\text{°C}$

$\begin{array}{l}\text{°X}\text{=}\frac{5}{14}(°C)-\frac{50}{14}\\ =\frac{5}{14}(22°C)-\frac{50}{14}\\ =4.2\text{°X}\end{array}$

Using above derived equation and substitute given temperature in Fig.1 $\text{22°C}$ to give a temperature in $\text{°C}$

Interpretation Introduction

Interpretation:

The relationship between $\text{°C}$ and $\text{°X}$ should be derived and the given $\text{°C}$ should be converted in to $\text{°X}$ and the given $\text{°X}$ should be converted in to degree Celsius, Kelvin and Fahrenheit.

Concept introduction:

Conversion of temperature units in between degree Celsius, Kelvin and Fahrenheit are given by

$\begin{array}{l}{\text{Conversion of }}^{\text{o}}\text{F}\text{into}{}^{\text{o}}\text{C:}\\ {}^{\text{o}}\text{C}\text{=}\left({}^{\text{o}}\text{F}\text{-}{\text{32}}^{\text{o}}\text{F}\right)\text{×}\frac{{\text{5}}^{\text{o}}\text{C}}{{\text{9}}^{\text{o}}\text{F}}\\ {\text{Conversion of }}^{\text{o}}\text{C}\text{into}{}^{\text{o}}\text{F:}\\ {}^{\text{o}}\text{F}\text{=}\left({}^{\text{o}}\text{C}\text{×}\frac{{\text{9}}^{\text{o}}\text{F}}{{\text{5}}^{\text{o}}\text{C}}\right)\text{+}{\text{32}}^{\text{o}}\text{F}\\ {\text{Conversion of }}^{\text{o}}\text{C}\text{into}\text{K:}\\ \text{K = °C + 273}\text{.15}\end{array}$

Answer to Problem 67E

The given $\text{58}\text{.0}\text{°X}$ converted in to degree Celsius, Kelvin and Fahrenheit are,

(i) $\text{52}\text{.4}\text{°C}$

(ii) $\text{452}\text{.22}\text{k}$

(iii) $\text{306}\text{.32}°\text{F}$

Explanation of Solution

The convert the given temperature $\text{58}\text{.0}\text{°X}$in to Celsius, Kelvin and Fahrenheit

$\begin{array}{l}\text{°C}\text{=}\frac{\text{14}}{\text{5}}\text{°X-10}\\ \text{°C}\text{=}\frac{\text{14}}{\text{5}}\text{58}\text{.0°X-10}\\ \text{=}\text{152}\text{.4}\text{°C}\\ \text{°F=}\left(\text{152}\text{.4°C×}\frac{\text{9°F}}{\text{5°C}}\right)\text{+32°F}\\ \text{=}\text{306}\text{.32°F}\\ \end{array}$

Using above derived equation and plugged the given values of given and calculated temperature to convert the temperatures in Kelvin and Fahrenheit unites.