A 1.00-L insulated bottle is full of tea at 88.0°C. You pour out a mug of tea and immediately screw the stopper back on the bottle. Find the change in temperature of the tea remaining in the bottle that results from the admission of air at room temperature. (Let the room temperature be 20.0°C and assume that you poured out 160 cm3 of tea. Take the molar mass of air as 28.9 g/mol and ρair = 1.20 10-3 g/cm3. Here we define a "monatomic ideal gas" to have molar specific heats CV = 3/2R and CP = 5/2 R, and a "diatomic ideal gas" to have CV = 5/2 R and CP = 7/2 R.)
To find the change in temperature of the tea remaining in the bottle, we can use the concept of thermal equilibrium and the principles of thermodynamics.
Step 1: Determine the initial temperature of the tea in the insulated bottle.
Given that the tea is initially at 88.0°C and the room temperature is 20.0°C, the initial temperature of the tea is 88.0°C.
Step 2: Calculate the final temperature of the tea after admission of air.
Since the tea is poured out and the stopper is screwed back on immediately, we can assume that no heat is lost or gained by the tea during the process. Therefore, the final temperature of the tea will be the same as the temperature of the room, which is 20.0°C.
Step 3: Determine the change in temperature of the tea.
ΔT = Tfinal - Tinitial
ΔT = 20.0°C - 88.0°C
ΔT = -68.0°C
Step 4: Convert the change in temperature from Celsius to Kelvin.
To convert from Celsius to Kelvin, we add 273.15 to the temperature in Celsius.
ΔT(K) = ΔT(°C) + 273.15
ΔT(K) = -68.0°C + 273.15
ΔT(K) = 205.15 K
Therefore, the change in temperature of the tea remaining in the bottle, resulting from the admission of air at room temperature, is -68.0°C or 205.15 K.
To find the change in temperature of the tea remaining in the bottle, we need to calculate the heat transfer that occurs when air at room temperature enters the insulated bottle.
First, let's calculate the mass of air that enters the bottle. We know that 160 cm³ of tea is poured out, and the density of air is 1.20 * 10⁻³ g/cm³. Therefore, the mass of air entering the bottle can be calculated as:
Mass of air = Volume * Density
= 160 cm³ * 1.20 * 10⁻³ g/cm³
= 0.192 g
Next, we need to calculate the heat transfer between the tea and the air, using the specific heat capacity of the air. To do this, we'll use the following formula:
Q = mcΔT
Where Q is the amount of heat transferred, m is the mass of the substance, c is the specific heat capacity, and ΔT is the change in temperature.
For monatomic ideal gas:
The specific heat capacity at constant volume (CV) is 3/2 R, and at constant pressure (CP) is 5/2 R.
For diatomic ideal gas:
The specific heat capacity at constant volume (CV) is 5/2 R, and at constant pressure (CP) is 7/2 R.
To determine which specific heat capacity to use, we need to identify whether air behaves like a monatomic or diatomic ideal gas.
The molar mass of air is given as 28.9 g/mol. Since air is a mixture of various gases, we'll assume that air behaves as a diatomic ideal gas. Therefore, we'll use the specific heat capacity values for a diatomic ideal gas.
Next, let's calculate the change in temperature. We'll assume that the tea and the air reach thermal equilibrium after mixing. The final temperature after mixing will be the same as the room temperature, which is 20.0°C.
Now we can calculate the change in temperature as:
ΔT = Tf - Ti
= 20.0°C - 88.0°C
= -68.0°C
Finally, we can calculate the heat transfer using the formula Q = mcΔT:
Q = (mass of air) * (specific heat capacity of air) * (change in temperature)
= 0.192 g * [(7/2) * R] * (-68.0°C)
Note: R is the ideal gas constant, which is 8.314 J/(mol·K)
Once you calculate Q, you can divide it by the amount of tea remaining in the bottle to determine the change in temperature of the tea.