At what temperature is the rms speed of helium molecules half its value at STP (0 C}, 1.0 atm)?

At what temperature is the rms speed of helium molecules twice its value at STP?

Isn't speed squared a measureof energy?

as is PV?

so if speed squared is measure of energy, then 1/2 speed is 1/4 energy, which means PV is 1/4, which means T is 1/4. Check my thinking.

I did'nt follow your approach

To find the temperature at which the root mean square (rms) speed of helium molecules is half its value at STP (Standard Temperature and Pressure: 0°C, 1.0 atm), we can make use of the relationship between temperature and rms speed.

The root mean square speed of gas molecules can be calculated using the following equation:

v = √[(3 * k * T) / (m * N)]

Where:
v - rms speed of gas molecules
k - Boltzmann constant (1.38 x 10^-23 J/K)
T - temperature (in Kelvin)
m - molar mass of helium (4 g/mol)
N - Avogadro's number (6.022 x 10^23 mol^-1)

To solve for the temperature at which the rms speed of helium molecules is half its value at STP, we can set up the following equation:

(v / 2)^2 = (3 * k * T) / (m * N)

Rearranging the equation, we get:

T = (m * N * (v / 2)^2) / (3 * k)

Substituting the known values:
m = 4 g/mol,
N = 6.022 x 10^23 mol^-1,
k = 1.38 x 10^-23 J/K,
v = rms speed at STP = ?

To find the rms speed at STP, we can use the ideal gas law:

PV = nRT

Where:
P = pressure (1.0 atm)
V = volume (1 mole of ideal gas occupies 22.414 L at STP)
n = number of moles (1 mole of helium)
R = ideal gas constant (0.0821 L.atm/mol.K)
T = temperature (STP: 0°C = 273.15 K)

Solving for V (volume):

V = (nRT) / P

Substituting the known values:
n = 1 mole,
R = 0.0821 L.atm/mol.K,
P = 1.0 atm,
T = 273.15 K,

V = (1 * 0.0821 * 273.15) / 1.0
V ≈ 22.414 L

Now, let's substitute the calculated volume into the equation for rms velocity:

v = √[(3 * k * T) / (m * N)]
v = √[(3 * (1.38 x 10^-23) * 273.15) / (4 * 6.022 x 10^23)]

By calculating this expression, you would find the rms speed of helium molecules at STP.

Finally, substitute the calculated rms velocity into the equation we derived earlier to find the temperature at which the rms speed is half its value at STP:

T = (m * N * (v / 2)^2) / (3 * k)

By substituting the known values for m, N, k, and v/2, you will obtain the temperature at which the rms speed of helium molecules is half its value at STP.