Let the mass of steam required to raise the temperature of 100 g of water from 24°C to 90°C be m gram of steam.
Each gram of steam on condensing releases 536 calories of heat. The steam which condenses is at 100°C, and it cools to final temperature of 90°C.
Heat released by m gram of steam on condensing= 536 × m calorie
Heat released by m gram of condensed steam condensed to water at 100°C to water at 90°C, the final temperature of the solution= m×specific heat of water× fall of temperature= m×1×10 = 10 m calories.
Total heat released by steam condensing and then cooling to 90°C = 536 m + 10 m = 546 m calories of heat.
Heat required to raise the temperature of 100 g of water at 24°C + m gram of condensed steam from 24°C to 90°C = (100 + m) ×1×(90°C - 24°C)= (100 + m)×66 calories
Using heat gained = Heat lost
(100 + m)×66 = 546 m; ==> 6600 + 66 m = 546 m; ====> 486 m = 6600, or m = 6600/480= 13.75 g of steam.
The amount of steam required is 13.75 g of steam. Remember one gram of water is about 1cc and it expands to about 1640 cc of steam at atmospheric pressure.