Whilst reading Hartshorne's appendix C I came across the comparison theorem for etale cohomology and singular cohomology:

Let $X$ be a smooth projective variety over a number field $K$ and $\ell$ a prime number. Fix an embedding of $K$ into $\mathbb C$. Then there is a "natural" isomorphism of $\mathbb C$-vector space $$\mathrm H^n(X_{\bar K,et},\mathbb Q_\ell)\otimes \mathbb C \cong \mathrm H^n(X(\mathbb C),\mathbb C).$$

Do we have the slightly stronger "natural" isomorphism of $\mathbb Q_\ell$-vector spaces

$$\mathrm H^n(X_{\bar K,et},\mathbb Q_\ell) \cong \mathrm H^n(X(\mathbb C),\mathbb Q)\otimes \mathbb Q_\ell?$$

I would appreciate references to the literature.