RMS speed is proportional to the square root of temperature. v_rms at 800 K = 300 * sqrt(800/400) = 300 * sqrt(2) = 600 m/s.

The kinetic energy per molecule is given by KE = 0.5 * m * v^2. Substituting v = 300 m/s gives the correct expression.

The RMS speed is proportional to the square root of the temperature. If the temperature is doubled, the RMS speed increases by a factor of sqrt(2). Therefore, the new RMS speed will be 300 * sqrt(2), which is approximately 600 m/s.

Using the formula v_rms = sqrt((3RT)/M), we can rearrange to find T: T = (M * v_rms^2) / (3R). Substituting M = 0.016 kg/mol and v_rms = 400 m/s gives T = 400 K.

The RMS speed increases with the square root of the temperature. Therefore, at 600 K, the RMS speed will be 400 * sqrt(2), which is approximately 800 m/s.

Kinetic energy per molecule = (1/2)mv^2. Using v = 400 m/s and m = M/N_A, we find KE = 0.5 mJ.

The average speed is related to RMS speed by the relation v_avg = (2/3) * v_rms. Thus, v_avg = (2/3) * 400 = 267 m/s.

The average speed of gas molecules is approximately 0.8 times the RMS speed, so average speed = 0.8 * 400 m/s = 320 m/s.

The speed at 1/2 of the RMS speed is simply 500 m/s / 2 = 250 m/s.

The average speed of gas molecules is related to the RMS speed by the relation v_avg = (v_rms * sqrt(8/3)). Therefore, the average speed is approximately 400 m/s.

The RMS speed is an average measure; individual molecular speeds will vary around this value.

Using v_rms = sqrt(3RT/M), we rearrange to find T = (v_rms^2 * M) / (3R). Plugging in values gives T approximately 300 K.

Using the sum of roots (-b/a = 3 + (-2) = 1), we find b = -1.

Using Vieta's formulas, a = 1 since the product of the roots (3 * -2) = -6 and sum (3 + -2) = 1.

The condition for equal roots is given by the discriminant: b^2 - 4ac = 0.

The discriminant is given by b^2 - 4ac. Here, b = 2, a = 1, c = 1, so the discriminant is 2^2 - 4*1*1 = 0.

The sum of the roots is -1 + -3 = -4, and the product is (-1)(-3) = 3. Thus, k = 3.

The sum of the roots is -b/a = -3/1 = -3.

Using Vieta's formulas, k = (-1)(-2) = 2.

For equal roots, the discriminant must be zero: 3^2 - 4*1*k = 0, hence k = 9.

For real and distinct roots, the discriminant must be positive: 3^2 - 4*1*k > 0 => 9 - 4k > 0 => k < 9.

The discriminant must be positive: 3^2 - 4*1*k > 0 leads to 9 - 4k > 0, thus k < 9.

Using the formula for the sum of roots, -2 + -2 = -4, and product of roots, (-2)(-2) = 4, we find k = 4.

For the roots to be equal, the discriminant must be zero. Thus, 4^2 - 4*1*k = 0 leads to k = 4.

The discriminant must be greater than zero: 4^2 - 4*1*k > 0 => 16 - 4k > 0 => k < 4.

For real and equal roots, the discriminant must be zero: 16 - 4k = 0, thus k = 4.

Using Vieta's formulas, the sum of the roots is -b/a = -5/1 = -5.

The product of the roots ab is given by c/a. Here, c = 6 and a = 1, so ab = 6.

Using the product of the roots, c = 2 * 3 = 6.

Using Vieta's formulas, k = (-2)(-3) = 6.