An ideal gas at 20 C consists of atoms 22*10^22. 4.5 J of thermal energy are removed from the gas. What is the new temperature in C?
The molar specific heat depends upon the type of gas. For monatomic gases it is 3/2 R or 2.98 Calories/mole*C. For diatomic gases (like O2 and N2) it is 4.97 Cal/mole*C. For polyatomic gaqses (like CO2) it is higher still.
You need to know what the gas is. Whoever assigned the problem should know that.
I assume that your number
22*10^22 is the number of molecules of gas. Divide that by Avogadro's number for the number of moles. It is about 1.38 moles.
Divide the Number of Calories removed by (molar specific heat)*(# of moles)
That will give you the temperature change. In this case it will be negative.
-272.97
To find the new temperature of the gas, we can use the formula:
ΔQ = nCΔT
Where:
ΔQ is the change in thermal energy,
n is the number of moles of the gas,
C is the molar specific heat capacity of the gas, and
ΔT is the change in temperature.
First, let's find the number of moles of the gas:
n = N / Avogadro's constant
Where:
N is the number of atoms, and
Avogadro's constant is approximately 6.022 x 10^23.
n = 22*10^22 / 6.022 x 10^23
n ≈ 0.0364 moles
Next, we need to find the molar specific heat capacity of the gas. Assuming it's an ideal gas at constant volume, the molar specific heat capacity (Cv) can be approximated as 3/2R, where R is the ideal gas constant.
Cv ≈ (3/2) * R
R is approximately 8.314 J/(mol*K), so we can substitute this value in:
Cv ≈ (3/2) * 8.314
Cv ≈ 12.471 J/(mol*K)
Now, we can rearrange the formula to solve for ΔT:
ΔT = ΔQ / (n * Cv)
ΔT = -4.5 J / (0.0364 moles * 12.471 J/(mol*K))
ΔT ≈ -10.87 K
Since the temperature is originally 20°C, we'll need to convert this to kelvin in order to add the change in temperature:
Initial temperature = 20°C + 273.15 ≈ 293.15 K
New temperature = 293.15 K + (-10.87 K)
New temperature ≈ 282.28 K
Finally, let's convert the new temperature back to Celsius:
New temperature = 282.28 K - 273.15
New temperature ≈ 9.13°C
Therefore, the new temperature of the gas is approximately 9.13°C.
To find the new temperature of the gas, we can make use of the First Law of Thermodynamics, which states that the change in internal energy of a system is equal to the heat added to or removed from the system, minus the work done by or on the system.
In this case, the internal energy of the gas is changing due to the removal of thermal energy. The change in energy (ΔU) can be calculated using the formula:
ΔU = q - w
where ΔU is the change in internal energy, q is the heat added or removed, and w is the work done.
Here, the heat removed from the gas is given as 4.5 J. Since no work is done on or by the gas (assuming it is an ideal gas), the work term can be neglected.
Therefore, ΔU = q = -4.5 J (as energy is being removed)
The change in internal energy of an ideal gas is related to its temperature change (ΔT) through the equation:
ΔU = nCvΔT
where n is the number of moles of gas, Cv is the molar specific heat capacity at constant volume, and ΔT is the change in temperature.
In this case, the number of moles (n) of the gas can be calculated using Avogadro's number (6.022 x 10^23 atoms/mol) and the given number of atoms:
n = 22 x 10^22 atoms / (6.022 x 10^23 atoms/mol)
= 0.0365 mol (approximately)
The molar specific heat capacity at constant volume for an ideal gas at constant volume can be considered as a constant (Cv = 3/2R), where R is the ideal gas constant (8.314 J/mol·K).
Plugging in the values, we have:
ΔU = nCvΔT
-4.5 J = (0.0365 mol)(3/2)(8.314 J/mol·K)ΔT
Solving for ΔT:
ΔT = (-4.5 J) / [(0.0365 mol)(3/2)(8.314 J/mol·K)]
≈ -67.8 K
Note: The negative sign indicates a decrease in temperature.
To convert this change in temperature to Celsius, we add the change to the initial temperature. The initial temperature is given as 20°C, which can be converted to Kelvin by adding 273.15.
New temperature in Kelvin = 20 + ΔT
= 20 - 67.8
≈ -47.8 °C
Therefore, the new temperature of the gas is approximately -47.8 °C.