Question

A scientist proposes a new temperature scale in which the ice point is 25X (X is the new unit of temperature) and the steam point is 305X. The specific heat capacity of water in this new scale in $Jk{{g}^{-1}}{{X}^{-1}}$, is:
A. $4\cdot 2\times {{10}^{3}}$
B. $3\cdot 0\times {{10}^{3}}$
C. $1\cdot 2\times {{10}^{3}}$
D. $1\cdot 5\times {{10}^{3}}$

Answer 1

Hint:Before we get to understand the measurement of temperature, it is important to know the definition of temperature.
Temperature is defined as the degree of hotness and coldness of a system, which primarily, indicates the direction of heat transfer between one or more thermodynamic systems.
Specific heat capacity is defined as the amount of heat required to raise the temperature of a unit mass of a substance through unit temperature.
The specific heat capacity of water in SI units is
$s=4200\dfrac{J}{kg{{-}^{\circ }}C}$

Complete step by step solution:
The temperature is different from heat. The heat is defined as the energy which is transferred across the system which can change the net total of the internal energy of the system whereas, the temperature is a quantity that manifests out of the heat energy, that measures the degree to decide the direction and the extent of transfer of heat energy from one system to another.

The temperature is measured by a device known as a thermometer. It is a device which contains a liquid such as alcohol or mercury is a small bulb, which expands into a small vertical capillary column, which has several graduation markings depending on the scale used to measure the temperature.

The graduations which are marked on the thermometer is based on a standard scale of measurement. Historically, there have been several scales of temperature, but the most common and the widely used scales of temperature are only 3: i) Celsius scale ii) Fahrenheit scale iii) Kelvin scale
As per this question, let us understand the new scale based on the Celsius scale.

The Celsius scale consists of 100 divisions starting from ${{0}^{\circ }}C$ which is the first reference point called the freezing point of water and the highest is the second reference point, the boiling point of water which is ${{100}^{\circ }}C$.
The difference between the standard points i.e. boiling point and freezing point in the Celsius scale = ${{100}^{\circ }}C-{{0}^{\circ }}C=100$
In the new scale,
The freezing point = 25X The boiling point = 305X
Here, X is the unit rise in temperature of the new scale.
Therefore, difference between the standard points i.e. boiling point and freezing point in the new scale:
$305X-25X=280X$
In the Celsius scale, the range between the ice point and steam point is divided into 100 equal parts. By correlating this with the new scale, we can say that the range of temperature in the new scale between ice point and steam point is divided into 280 parts.
Hence, one-degree change in Celsius scale corresponds to a change in the new scale, equal to –
$X=\dfrac{280}{100}=2\cdot {{8}^{\circ }}C$
This equivalent change should be substituted in the formula for specific heat capacity to obtain the answer.
Specific heat capacity of water, $s=4200\dfrac{J}{kg{{-}^{\circ }}C}$
Here, in the place of one-degree Celsius, let us substitute the corresponding change in the new scale.
Specific heat capacity of water, $s=\dfrac{4200}{2\cdot 8}\dfrac{J}{kg-X}=1\cdot 5\times {{10}^{3}}Jk{{g}^{-1}}{{X}^{-1}}$

Hence, the correct option is Option D.

Note:By applying the similar concept mentioned here, for the Fahrenheit scale, where the ice point is ${{32}^{\circ }}F$ and the steam point is ${{212}^{\circ }}F$, the unit rise in Fahrenheit scale corresponding to the unit change in Celsius scale is –
$\Rightarrow \dfrac{212-32}{100}=\dfrac{180}{100}=1\cdot {{8}^{\circ }}F$
This proves that every ${{1}^{\circ }}C$ rise in the Celsius scale corresponds to $1\cdot {{8}^{\circ }}F$ rise in Fahrenheit scale.