Final answer:
The temperature at which one mole of diatomic oxygen has 5.0 x 10^3 J of kinetic energy is approximately -32.15 °C, calculated using the relationship between kinetic energy and temperature for gases.
Step-by-step explanation:
In physics, especially thermodynamics, the kinetic energy of gas molecules is related to temperature through the equipartition theorem.
According to this theorem, for a diatomic gas such as oxygen (O2), each molecule has 5 degrees of freedom: 3 translational and 2 rotational, because the vibrational modes only come into play at much higher temperatures.
The total kinetic energy (KE) of one mole of a diatomic gas is given by KE = (5/2)nRT, where n is the number of moles, R is the universal gas constant (8.314 J/mol·K), and T is the temperature in kelvins (K).
Given that one mole of diatomic oxygen has 5.0 x 103 J of kinetic energy, we can calculate the temperature.
Setting up the equation 5.0 x 103 J = (5/2)(1 mol)(8.314 J/mol·K)T,
we solve for T.
Dividing both sides of the equation by (5/2)(8.314) gives us T ≈ 241 K.
To convert this to Celsius (°C), we use the conversion formula C = K - 273.15, which results in a temperature of approximately -32.15 °C.