To determine the final temperature when a liter of water at 10°C is sprayed into 10 cubic meters of air at 90°C, we can use the principle of energy conservation.
First, we need to calculate the initial energy of both the water and the air.
For water: Initial energy = mass of water × specific heat capacity of water × change in temperature = 1 kg × 4186 J/(kg·°C) × (final temperature - 10°C)
For air: Initial energy = mass of air × specific heat capacity of air × change in temperature = density of air × volume of air × specific heat capacity of air × (final temperature - 90°C)
Since we're given the volume of air (10 cubic meters) and its initial temperature (90°C), we need to determine the mass of air using the ideal gas law.
The ideal gas law states: PV = nRT
Assuming air behaves ideally, we can rearrange the equation as follows: n = PV / RT
Given: P = atmospheric pressure (assume standard pressure, approximately 101325 Pa) V = volume of air (10 cubic meters) R = ideal gas constant (approximately 8.314 J/(mol·K)) T = initial temperature of air (90°C + 273.15 K)
Now we can calculate the mass of air using the molar mass of air (approximately 28.97 g/mol): mass of air = (PV / RT) × molar mass of air
Next, we equate the initial energy of water and air, and solve for the final temperature: 1 kg × 4186 J/(kg·°C) × (final temperature - 10°C) = mass of air × specific heat capacity of air × (final temperature - 90°C)
Rearranging the equation and substituting the values, we can solve for the final temperature.
However, please note that this calculation assumes ideal behavior for air and neglects any heat loss to the surroundings. In reality, factors like heat transfer, humidity, and specific conditions may affect the actual final temperature.