# Question about model proposed by Boltzmann for the ideal gas

The following text is from the book Introdução à Física Estatística (Introduction to Statistical Physics), Silvio Salinas. There's something in it that i didn't understand.

In order to highlight the role of probabilistic knowledge in defining entropy, Boltzmann proposed a model in which the particles of a gas could be found in a discrete set of energy values, {$$\epsilon_j ; j = 1,2,3...$$}. The microscopic state of the Boltzmann gas is specified by supplying the energy of each of the particles. So, let's define the set of "occupation numbers", {$$N_j$$}, Where $$N_1$$ designates the number of particles with energy $$\epsilon_1$$, $$N_2$$ designates the number of particles with energy $$\epsilon_2$$, and so on. Therefore, given the set of occupation numbers {$$N_j$$}, the total energy E and the total number of particles N, we can write in the total number of microstates accessible to the system in the form $$\Omega(\{N_j\},E,N) = \frac{N!}{N_1!N_2!N_3!...} \tag1$$ with the constraints $$N = \sum_{j} N_j \quad \text{and} \quad E=\sum_{j}\epsilon_j N_j \tag2$$ the probability of finding the system in these conditions will be proportional to $$\Omega (\{ N_j \},E,N)$$, with the constraints imposed by the equations (2). So in order to find the occupation numbers in equilibrium, we can maximize $$\Omega$$ with respect to the set {$$N_1$$}. Using Lagrange multipliers method, we have the function $$f(\{ N_j \}, \lambda_1, \lambda_2) = \log \Omega (\{ N_j \}, E, N)) + \lambda_1 (N - \sum_{j} N_j) + \lambda_2 (E - \sum_{j} \epsilon_j N_J). \tag3$$

Using Stirling's Approximation, we can also write in $$\frac{\partial f}{\partial N_k} = - \log N_k - \lambda_1 - \lambda_2 \epsilon_k = 0 \tag4$$

Eliminating the Lagrange's multiplier $$\lambda_1$$, we have the equilibrium distribution $$\frac{N_k}{N} = \frac{\exp(- \lambda_2 \epsilon_k)}{Z_1} \tag5$$ where the normalization factor $$Z_1$$ is defined as $$Z_1 \equiv \sum_{j} \exp(- \lambda_2 \epsilon_j) \tag6$$

I didn't understand how the equation (5) was obtained, that is, how that $$\lambda_1$$ was eliminated in order to obtain equation (5).

We use the Lagrange multiplier $$\lambda_1$$ to enforce the constraint $$N - \sum_j N_j = 0$$, that is, total number of particles sums to $$N$$. Solving (4) yields
$$N_k = e^{-\lambda_1} e^{-\lambda_2 \epsilon_k}$$
$$N = \sum_k N_k = e^{-\lambda_1} \sum_k e^{-\lambda_2 \epsilon_k}$$
so that means that $$e^{-\lambda_1} = \frac{N}{Z_1}$$, using the normalization factor defined in (6).