Answer the following:
$(a)$ The triple-point of water is a standard fixed point in modern thermometry. Why? What is wrong with taking the melting point of ice and the boiling point of water as standard fixed points (as was originally done in the Celsius scale)?
$(b)$ There were two fixed points in the original Celsius scale as mentioned above,which were assigned the numbers $0^{\circ}C$ and $100^{\circ}C$ respectively. On the absolute scale,one of the fixed points is the triple-point of water,which on the Kelvin absolute scale is assigned the number $273.16\; K$. What is the other fixed point on this (Kelvin) scale?
$(c)$ The absolute temperature (Kelvin scale) $T$ is related to the temperature $t_c$ on the Celsius scale by $t_c = T - 273.15$. Why do we have $273.15$ in this relation,and not $273.16$?
$(d)$ What is the temperature of the triple-point of water on an absolute scale whose unit interval size is equal to that of the Fahrenheit scale?

(N/A) The triple-point of water is a unique state that occurs at a specific temperature and pressure,making it a highly reproducible standard. The melting point of ice and the boiling point of water are dependent on pressure; therefore,they are not unique unless the pressure is strictly defined,making them less reliable as universal standards.
$(b)$ The other fixed point on the Kelvin absolute scale is absolute zero,which is $0\; K$.
$(c)$ The value $273.16\; K$ is the triple-point of water. The Celsius scale defines $0^{\circ}C$ as the melting point of ice at $1\; atm$ pressure. The difference between the triple-point of water and the melting point of ice is $0.01\; K$. Thus,$0^{\circ}C = 273.16\; K - 0.01\; K = 273.15\; K$. Hence,$t_c = T - 273.15$.
$(d)$ The size of $1\; K$ is equal to $1.8$ times the size of $1^{\circ}F$ (since $100\; K$ interval corresponds to $180^{\circ}F$ interval). The triple-point of water is $273.16\; K$. On a scale where the unit interval is equal to the Fahrenheit scale,the triple-point temperature is $273.16 \times 1.8 = 491.688\; R$ (Rankine scale).