This question involves analyzing the behavior of an ideal gas using a plot of Pd versus P, where P is pressure and d is density. We are given a derivative condition and need to find the temperature.

Step 1: Express density in terms of ideal gas variables

For an ideal gas, PV = nRT. Density d = mass/volume = m/V. Also, number of moles n = m/M, where M is molar mass.

So, PV = (m/M)RT ⇒ V = (mRT)/(PM)

Density d = m/V = m / (mRT/PM) = (PM)/(RT)

Thus, d = (PM)/(RT) ...(1)

Step 2: Form the product Pd

Multiply both sides of (1) by P:

Pd = P * (PM)/(RT) = (P²M)/(RT) ...(2)

Step 3: Differentiate Pd with respect to P

From (2), Pd = (M/(RT)) * P²

Let k = M/(RT), which is constant at fixed T.

So, Pd = k P²

Now, differentiate with respect to P:

d(Pd)/dP = d(kP²)/dP = 2kP = 2*(M/(RT))*P ...(3)

Step 4: Use the given condition

Given at P = 8.21 atm, d(Pd)/dP = 10

So, from (3): 10 = 2 * (M/(RT)) * 8.21

For H₂ gas, molar mass M = 2 g/mol

So, 10 = 2 * (2/(R T)) * 8.21

Simplify: 10 = (4 * 8.21) / (R T) = 32.84 / (R T)

So, R T = 32.84 / 10 = 3.284

Gas constant R = 0.0821 L·atm·mol⁻¹·K⁻¹ (since pressure is in atm and density in gm/L, consistent units)

So, 0.0821 * T = 3.284

Thus, T = 3.284 / 0.0821 = 40 K

Final Answer: 40 K

Related Topics

Ideal Gas Law: The equation PV = nRT describes the behavior of an ideal gas, relating pressure, volume, temperature, and number of moles.

Density of Gases: For an ideal gas, density d = PM/(RT), which is derived from the ideal gas law by substituting mass and molar mass.

Differentiation in Physical Chemistry Calculus is often used to find rates of change or slopes in gas law problems, especially when dealing with plots involving variables like P, V, T.

Formulae

Ideal Gas Law: $PV=nRT$

Density: $d=\frac{PM}{RT}$

Product Pd: $Pd=\frac{P^{2}M}{RT}$

Derivative: $\frac{d}{dP}\left(Pd\right)=\frac{2PM}{RT}$