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When water is heated from $0^{\circ} C$ to $4^{\circ} C$ and $C_{p}$ and $C_{v}$ are its specific heated at constant pressure and constant volume respectively, then:
A.${{C}_{p}}>{{C}_{v}}$
B.${{C}_{p}}<{{C}_{v}}$
C.${{C}_{p}}={{C}_{v}}$
D.${{C}_{p}}-{{C}_{v}}=R$

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Last updated date: 15th Jun 2024
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Answer
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Hint: The molar heat capacity is represented by $C_p$, and when pressure is constant, C. In other words, without any change in the volume of that system, $C_v$ is the heat energy transfer between a system and its surroundings. When volume is constant, $C_v$ represents the molar heat capacity C. Calculate water density and the volume of the water and then integrate $C_p$ with respect to time. Integrate $C_v$ with respect to time Equate both the equations.

Complete solution Step-by-Step:
The value of the specific heat capacity ($C_p$) for liquids at room temperature and pressure is approximately 4.2 J/g°C. This implies that raising 1 gramme of water by 1 degree Celsius requires 4.2 joules of energy. This value is actually quite large for $C_p$. The heat of the body rises as the body absorbs heat, but it cools down when heat is withdrawn from the body, so the heat of the body decreases. The temperature of any body is the measure of the kinetic energy of its molecules. The ratio of heat absorbed by a material to the temperature change is heat capacity.

Water has the highest density at $4^{\circ} \mathrm{C}$. This changes its properties from other simple fluids.

When water is heated from $0^{\circ} \mathrm{C}$ to $4^{\circ} \mathrm{C}$, the volume of liquid decreases. Thus, for this transition, $\mathrm{P} \Delta \mathrm{V}$ is negative.
\[\begin{array}{*{35}{l}}
   \int{{{\text{C}}_{\text{P}}}}\text{dT}=\int{{{\text{C}}_{\text{V}}}}\text{dT}+\text{P}\Delta \text{V} \\
   {} \\
\end{array}\]
$\therefore {{\text{C}}_{\text{P}}}<{{\text{C}}_{\text{V}}}$

So, When water is heated from $0^{\circ} C$ to $4^{\circ} C$ and $C_{p}$ and $C_{v}$ are its specific heated at constant pressure and constant volume respectively, then: ${{\text{C}}_{\text{P}}}<{{\text{C}}_{\text{V}}}$

Hence, the correct option is B.

Note:
The ratio of heat capacity at constant pressure ($C_P$) to heat capacity at constant volume ($C_V$) is the ratio of heat capacity at constant pressure ($C_P$) to heat capacity at constant volume ($C_V$) in thermal physics and thermodynamics, also known as the adiabatic index, the ratio of specific heat, or Laplace's coefficient. The pressure inside is equal to that of the atmosphere.