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Arginine vasopressin is a pituitary hormone. It helps to regulate the amount of water in the blood by reducing the flow of urine from the kidneys. An aqueous solution containing 21.6 mg of vasopressin in 100 mL of solution had an osmotic pressure of 3.70 mm Hg at 25$^{0}C$. What is the molecular weight of the hormone?

Answer
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Hint: Antidiuretic hormone also known as vasopressin is a hormone which is synthesized as a peptide prohormone in the neurons present in the hypothalamus and it gets converted into arginine vasopressin (AVP).

Complete answer:
Given in the question:
Mass of vasopressin = 21.6 mg
Mass of vasopressin (w)= $(21.6)({{10}^{-3}})$ g
The volume of the solution is = 100 ml
Volume of the solution (V)= $\dfrac{100}{1000}$ liters
Osmotic pressure of the solution = 3.70 mm
Osmotic pressure of the solution ($\pi $) =$\dfrac{3.70}{760}$ atm
Temperature of the solution = 25 $^{0}C$
Temperature of the solution (T) = $25+273$= 298 K
Now we use have to find the molecular weight we will use the given formulae which gives a relation between all the given quantities:
\[\pi V=\dfrac{w}{M}RT\]
\[(\dfrac{3.70}{760})(\dfrac{100}{1000})=(\dfrac{(21.6)({{10}^{-3}})}{M})(0.0821)(298)\]
Hence, an aqueous solution containing 21.6 mg of vasopressin in 100 mL of solution had an osmotic pressure of 3.70 mm Hg at 25$^{0}C$the molecular weight of the hormone (M) = 1085 g/mol

Additional information:
AVP has two important functions:
It increases the amount of solute free water which gets reabsorbed back into the circulation from the filtrate in the nephrons of the kidney tubules.
AVP will constrict the arterioles, which will eventually raise the pressure in the arteries. If the AVP gets released in high concentration then it will raise the blood pressure by inducing moderate vasoconstriction.

Note:
Some of the AVP get secreted directly to the brain from the hypothalamus and it plays an important role in social behavior. It is also responsible for the maternal response to stress and promotes heart muscles homeostasis.