Set ΔG = 0: 0 = ΔH - TΔS; T = ΔH/ΔS = 50 kJ / 0.1 kJ/K = 500 K.

The Gibbs Free Energy changes from negative to positive at the transition temperature, where the system shifts from one phase to another.

According to Charles's Law, the volume of a gas approaches zero at absolute zero, which is -273.15 °C.

According to Charles's Law, the volume of a gas approaches zero at absolute zero, which is 0 K.

Using v_rms = sqrt(3RT/M), we rearrange to find T = (v_rms^2 * M) / (3R). Plugging in values gives T approximately 500 K.

Using v_rms = sqrt(3RT/M), we solve for T: T = (v_rms^2 * M) / (3R) = (1000^2 * 0.044) / (3 * 8.314) = 700 K.

Using the formula v_rms = sqrt((3RT)/M), we can rearrange to find T. Setting v_rms = 300 m/s and M = 28 g/mol, we find T = (M * v_rms^2)/(3R) = 600 K.

Using v_rms = sqrt(3RT/M), rearranging gives T = (v_rms^2 * M) / (3R). Substituting values gives T = 500 K.

Using v_rms = sqrt(3RT/M), we can rearrange to find T = (v_rms^2 * M) / (3R). Plugging in values gives T = (600^2 * 0.02) / (3 * 8.314) = 900 K.

The area is given by the integral from 0 to 1 of (x - x^2) dx. This evaluates to [x^2/2 - x^3/3] from 0 to 1 = (1/2 - 1/3) = 1/6 = 0.5.

The area is given by the integral from 0 to 2 of (2x - x^2) dx. This evaluates to [x^2 - x^3/3] from 0 to 2 = (4 - 8/3) = 4/3.

The area is given by the integral from 0 to 2 of (4 - x^2) dx. This evaluates to [4x - x^3/3] from 0 to 2 = (8 - 8/3) = 16/3.

The area under the curve y = cos(x) from x = 0 to x = π/2 is given by ∫(from 0 to π/2) cos(x) dx = [sin(x)] from 0 to π/2 = 1 - 0 = 1.

The area under the curve is given by ∫(from 0 to 2) (x^2 + 2x) dx = [x^3/3 + x^2] from 0 to 2 = (8/3 + 4) = 20/3.

The area under the curve y = x^4 from x = 0 to x = 2 is given by ∫(from 0 to 2) x^4 dx = [x^5/5] from 0 to 2 = (32/5) - 0 = 32/5.

Using the chain rule, f'(x) = d/dx(e^(2x)) = 2e^(2x).

Using the product rule, f'(x) = d/dx(x^2 * e^x) = (x^2 + 2x)e^x.

Determinant = (1*4) - (2*3) = 4 - 6 = -2.

The determinant is calculated as \( 2*7 - 3*5 = 14 - 15 = -1 \).

The determinant is calculated as \( 2(0*2 - 1*1) - 1(1*2 - 3*1) + 3(1*1 - 3*0) = 0 \).

The determinant is calculated as \( 2*4 - 3*1 = 8 - 3 = 5 \).

The determinant is calculated as (2*7) - (3*5) = 14 - 15 = -1.

The rows are linearly dependent, hence the determinant is 0.

The determinant is 0 because the rows are linearly dependent.

The determinant is \( 2*7 - 3*5 = 14 - 15 = -1 \).

The determinant is calculated as \( ad - bc \).

The determinant of the identity matrix is 1.

The first determinant is -2 and the second is 0, so the total is -2 + 0 = -2.

The determinant evaluates to 0 as the rows are linearly dependent.

Using the determinant formula, we find that the determinant evaluates to 0.