MHB Maths in Temperature and ideal gas Thermometer

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The discussion centers on understanding the mathematics related to temperature measurements using an ideal gas thermometer. It clarifies that the triple point of water is defined as 273.16 K, while the freezing point of water is 273.15 K at 1 atm, leading to a distinction between 0°C and the triple point. The subscript "P" in the mathematical expression $\big(\frac{\partial V}{\partial T}\big)_P$ indicates that the derivative is taken at constant pressure. The relationship for an ideal gas shows that volume changes with temperature while pressure remains constant, resulting in the derivative being $\frac{R}{P}$. This explanation aims to resolve confusion about the definitions and mathematical expressions involved.
WMDhamnekar
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Hi,
I didn't understand the maths involved in the below article in regard to temperature and ideal gas thermometer. If any member knows it, may reply me.

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If triple point of water is fixed at 273.16 K, and experiments show that freezing point of air-saturated water is 273.15 K at 1 atm system pressure.(so, melting point of ice is also 273.15 K , then how triple point of water is $0.10^\circ C$ What is meant by real gas volume at thermal equilibrium with a system whose true temperature is $V_T$ be V. In this Math symbol$\big(\frac{\partial V}{\partial T}\big)_P$ what does subscript P indicate? My guess P means Partial.

I am stuck here. If get the answers to my questions satisfactorily, i shall proceed further.
 

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Dhamnekar Winod said:
If triple point of water is fixed at 273.16 K, and experiments show that freezing point of air-saturated water is 273.15 K at 1 atm system pressure.(so, melting point of ice is also 273.15 K , then how triple point of water is $0.10^\circ C$

$0\,^\circ C$ is defined to be the freezing point of water at the standard 1 atm pressure.
It corresponds to $273.15\,K$.
Since the triple point of water is at $273.16\,K$, then that means that it corresponds to $0.01\,^\circ C$.

What is meant by real gas volume at thermal equilibrium with a system whose true temperature is $V_T$ be V. In this Math symbol$\big(\frac{\partial V}{\partial T}\big)_P$ what does subscript P indicate? My guess P means Partial.

I am stuck here. If get the answers to my questions satisfactorily, i shall proceed further.
The subscript $P$ in the expression $\big(\frac{\partial V}{\partial T}\big)_P$ stands for the pressure $P$.
It means that we take the derivative of $V$ with respect to $T$ while we keep the pressure $P$ constant.

We have $V=\frac{RT}{P}$ for an ideal gas.
When we take the derivative, we treat $P$ as a constant so that $\big(\frac{\partial V}{\partial T}\big)_P=\big(\frac{\partial}{\partial T}\frac{RT}{P}\big)_P=\frac{R}{P}$.
 
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