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# Science

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German physicist Max Planck built a theory of black body radiation and used it to help start the field of quantum mechanics.

Planck's theory predicted:$$f_{\rm peak} = 2.821 \dfrac{k_B T}{h}.$$

In this formula, $$f_{\rm peak}$$ is the peak frequency of radiation coming from a black body like a star or the one in the experiment we've been analyzing.  is a constant from thermodynamics that allows scientists to convert from temperatures to energies. In Planck's time, it was known to be$$k_B \approx 1.38 \times 10^{-23} \;\mathrm{m^2 \cdot kg \cdot s^{-2} \cdot K^{-1}}.$$

Based on your analysis of the temperature and frequency data, what value does this experiment give for h?

$$h= ? \times 10^? kg^? m^? s^?$$

Apr 16, 2024

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Absolutely, based on the formula and the constants provided, we can calculate the value of Planck's constant (h) from the experimental data.

Here's how:

Identify the Given Values:

f_peak (peak frequency): This value likely comes from your experiment and needs to be plugged in. We'll denote it as an unknown for now.

k_B (Boltzmann's constant): k_B ≈ 1.38 × 10^-23 m² kg s⁻² K⁻¹ (given)

T (temperature): This value likely comes from your experiment and needs to be plugged in. We'll denote it as an unknown for now.

Rearrange the Formula for h:

We want to isolate h on one side of the equation. The formula is:

f_peak = 2.821 * (k_B * T) / h

To solve for h, multiply both sides by h and divide both sides by 2.821 * k_B * T:

h = (f_peak * 2.821 * k_B * T) / (f_peak)

We can simplify this further since f_peak appears in both the numerator and denominator and cancels out:

h = 2.821 * k_B * T

Plug in Experimental Data:

Now, replace f_peak and T with the actual values you obtained from your experiment. Make sure the units are consistent.

h = 2.821 * (1.38 × 10^-23 m² kg s⁻² K⁻¹) * T (in Kelvin)

Note: Since k_B is a very small number, the final value of h will depend significantly on the measured values of f_peak and T.

Example:

Let's say your experiment measured a peak frequency of f_peak = 5.00 x 10^14 Hz (hertz) and the temperature of the black body was T = 6000 Kelvin.

h = 2.821 * (1.38 × 10^-23 m² kg s⁻² K⁻¹) * 6000 K

h ≈ 6.61 x 10^-34 J s (joules per second)

This is a typical range for the value of Planck's constant.

Apr 17, 2024